IMPORTANT:

Achievements cannot be earned while playing with ANY mods. Whether it is a mod with a significant impact on gameplay or just a texture replacer, it will disable achievements for the entirety of that game

IMPORTANT:

Choosing "Restart Game" from the options after starting a game will also disable achievements for that game. If you want to get a new seed on your start, you must return to the main menu and setup a new game altogether.

Chief Extraterrestrial Officer - Win a game as Fielding (ARC)
The Big Payoff - Win a game as Hutama (Polystralia)
Shining Path - Win a game as Daoming (PAC)
Gagarin's Legacy - Win a game as Kozlov (Slavic Federation)
Father of Nations - Win a game as Barre (PAU)
The Art of the Infinite - Win a game as Élodie (Franco-Iberia)
Never Surrender - Win a game as Bolivar (Brasilia)
Moksha - Win a game as Kavitha (Kavithan Protectorate)
Grand Galactic Inquisitor - Win at least one game with each leader

The leader achievements are fairly straightforward. Simply win a separate game with each of the 8 leaders. This can be done on any difficulty, any map size, any number of players, and with an victory condition.

The fastest way to win a game is with the Coup rush explained later for the difficulty achievements.

For efficiency's sake, it makes the most sense to complete these achievements, the map size achievements, the map type achievements, and the victory condition achievements simultaneously. Example: Win as Bolivar on a Standard size Archipelago map with the Promised Land victory conditon.

Beep… Beep…. Beep... - Win a game on Sputnik Difficulty
Godspeed - Win a game on Mercury Difficulty
Поехали! (Pojexali!) - Win a game on Vostok Difficulty
8 Days or Bust - Win a game on Gemini Difficulty
Order of Lenin - Win a game on Soyuz Difficulty
The Eagle Has Landed - Win a game on Apollo Difficulty

The difficulty achievements will likely be the greatest challenge. While anyone with a small grasp of the game mechanics should be able to win on Sputnik and anyone with a moderate amount of experience can win on Mercury, the other difficulties get progressively more challenging up to Soyuz followed by a large jump to Apollo.

While I personally recommend attempting to earn these achievements through typical gameplay, everyone has different goals and skillsets. If you reach a difficulty you can't overcome, but still want the achievement, there is a simple way to cheese a victory on any difficulty.

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RUSH COUP CHEESE METHOD:

This method uses the level 5 espionage ability Coup D'etat to steal a city from another player. During the mid-late game, most AIs will make this very difficult by maintaining their Health to keep Intrigue down and by building Surveillance Webs to reduce the maximum Intrigue level, effectively preventing other players from using high level espionage. In the early game, however, there is no real protection.

SETUP: For this method, begin a game on any map size and type with only a single AI opponent on Quick. Play as ARC for the espionage bonus and with Scientists for the beakers. This will work against any opponent, but I prefer Kavitha because she seems friendlier in the early game, but any opponent will be unlikely to rush military early if you don't start as neighbors. The other options don't matter as much. If you aren't able to successfully stage a Coup D'etat, choosing to start with Pioneering instead of a Worker won't make a difference. I usually take Machinery and Tectonic Scanner anyway though.

UNITS: After placing your capital, start exploring with your Explorer for Resource Pods and expeditions. Even if you don't find any, you'll be okay, but finding a Science bonus is definitely nice. If you chose Machinery, set your Worker to work whatever. You won't need any military or colonists for this strategy.

TECH TREE: For your tech choice, rush immediately to Computing through Ecology. Everything else you do matters very little. After you have Computing, if your explorer is still alive, you may want to go for Genetic Mapping to get the Purity Level. But as I said, Computing is the only important tech.

PRODUCTION: For your production, start off by building an Old Earth Relic for the culture bonus and then prioritze Research to get to Computing as fast as possible. Once you have Computing, get your Spy Agency built immediately. Force your citizens to work your highest production tiles to speed it up. Once you have your Spy Agency built, it no longer matters what you do.

VIRTUES: For your virtues, you'll want to rush to Special Service, which is a Tier 2 virtue in the Might tree. Go Adaptive Tactics -> Military Industrial Complex -> Liberation Army -> Special Service. If you built an Old Earth Relic, you'll get plenty of culture to reach Special Service before you start your first Coup D'etat. After that, it doesn't matter what you do.

ESPIONAGE: Here's how you win. Once you finish building your Spy Agency, choose the Operations Reconnaissance project and assign one spy to your opponent's capital. Leave the rest at your HQ. When he arrives, Establish a Network and then choose the highest available level op each time to build up Intrigue. Eventually, you'll have access to Coup D'etat. It may take more than one attempt, but when it succeeds, you'll win a Domination Victory.

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Resistance is Futile - Achieve Domination Victory

This victory condition means being the last civilization in control of your original capital. It is important to note this does NOT mean you have to control every capital. You only have to be the only civ who controls their own capital. This is why when a civ loses their capital, every other civ gets the same victory condition progress. The Rush Coup method wins with this method very quickly on any difficulty.

We Are Not Alone - Achieve Contact Victory

Contact is a peaceful victory not associated with any affinity. To earn this victory, you need to complete two of the following:

Find The Signal from a Progenitor Ruins with an Explorer (Rare outcome)
Launch a Deep Space Telescope (From Orbital Automation tech, requires 2 Petroleum and 1 Firaxite)
Build the Transcendental Equation (National Wonder from Transcendental Math tech)

And then build the Decode Signal Wonder followed by the Contact Beacon wonder. After that, select the Contact Beacon and activate it by spending 1000 Energy and then wait the specified number of turns. All of your Energy per Turn will go to speeding it up.

Phone Home - Achieve Promised land

This is the Purity victory. To earn this victory you must launch a Lasercom Satellite, which costs 2 Petroleum and 1 Titanium and is unlocked from the Orbital Networks tech. Then reach Level 13 Purity and build the Exodus Gate, unlocked from the Nanotechnology tech. Once it is built, once per turn, you can select it and summon in a Colonist from Earth. To finish, you'll need to settle 20 Earth Colonists safely. Settlements cannot be made within 3 tiles of another city, but after one is made, 5 more colonists can settle there making a minimum of 4 settlements to win. To simplify things, you may consider spacing your cities out more than you would normally to make room for settlements that don't require putting your colonists at risk.

Planned Obsolescence - Achieve Emancipation

This is the Supremacy victory. To earn this victory you must launch a Lasercom Satellite, which costs 2 Petroleum and 1 Titanium and is unlocked from the Orbital Networks tech. Then reach Level 13 Supremacy and build the Emancipation Gate, unlocked from the Hypercomputing tech. Once it is built, you must send land units to it totalling 1000 Strength. Only one unit per turn can be sent through the gate.

It's Full of Stars - Achieve Transcendence

This is the Harmony Victory. To earn this victory, Research Nanorobotics, Transgenics, and Swarm Intelligence and reach Level 13 Harmony. Doing so will unlock the Mind Flower wonder. Once it is built, all aliens on the map will become aggressive regardless of their current level of hostility. You'll win a Transcendence victory after waiting so many turns after the Mind Flower is completed. Your turn total is lowered for each Mind Stem and Xeno Sancturary you built PRIOR to the Mind Flower.

Steely Eyed Missile Man - Achieve one of each other victory

Straightforward achievement here. Just finish a game with each of the previously explained conditions. If you choose "One more turn..." after winning a game, you can continue playing and earn a different victory achievement.

Aim to Misbehave - Win a game on a Duel Map
Tiny Big Planet - Win a game on a Tiny Map
That's No Moon - Win a game on a Small Map
Pale Blue Dot - Win a game on a Standard Map
The View is Tremendous - Win a game on a Large Map

These can be earned on any difficulty with any victory condition and with any number of players. I can't explain why they're different, but in-game Tiny is Dwarf and Large is Massive.

Poseidon's Children - Win game on Atlantean Map
That New Planet Smell - Win game on Protean Map
A Planet the Very Brother of Your Own - Win game on Terran Map
When Maui Came To This World - Win game on Archipelago Map
Just Like Voskhod 2 - Win game on Taigan Map
Cruel and Unusual Geography - Win game on Equatorial Map
Fearful Symmetry - Win game on Skirmish Map

These can be earned on any difficulty with any victory condition and with any number of players.

United Federation of Planets - Win at least one game on each map size and type

You'll earn this achievement after winning a game on all 5 map sizes and all 7 map types.

Cylon Computer Virus - Win a multiplayer game

You'll get credit for this achievement in a Hot Seat game as well as Online Mutiplayer.

The Machineries of Joy - Build any wonder

As it says, build absolutely ANY wonder to earn this achievement.

I've made a lot of special modifications myself - Download a mod

Go to the Steam Workshop and subscribe to any mod. Then in game, choose Mods from the main menu and you'll earn this achievement when your first mod finishes downloading.

I'm in the middle of some calibrations - Play with a Mod

This is the exception to the no mods rule. You'll earn this achievement after starting a game with any mod. You can immediately quit the game after it begins.

I'm On Another Boat - Embark a unit

Simply reseach Planetary Survey and order a unit to step into a water tile.

Game over, man! - One Hit Kill

Most commonly achieved against an alien in mid to late game with a Tier 3+ Unit.

Mark II - Upgrade a Unit

Reach Level 1 of any Affinity to upgrade a Soldier to a Marine.

Enemy Within - Kill an enemy spy

After building the Spy Agency, have your spies travel to your cities to defend them against enemy espionage. Eventually an enemy spy will be killed without you having done anything else.

Patent Pending - Steal a tech

After building the Spy Agency, have your spies travel to enemy cities and perform ops to build Intrigue until the Steal Technology Op is unlocked. Complete that covert op to earn this achievement.

Light This Candle - Launch an orbital unit

You'll most likely earn this achievement by launching a free Solar Collector courtesy of your Explorer, but, if not, building and launching any satellite yourself will also work.

Valley of the Time Tombs - Build all wonders

Build every possible wonder across any number of games. You can see which wonders you are missing through the following steps:

Go to "My Documents\My Games\Sid Meier's Civiliztion Beyond Earth"
Open "config.ini"
Change "DebugPanel = 0" to "DebugPanel = 1"
Go to "My Documents\My Games\Sid Meier's Civiliztion Beyond Earth\Logs"
Open "achievement_debug.log"

At the bottom you'll see how many of each wonder you have built across all games. The wonders marked with a 0 have never been built.

Mighty Fine Shindig - Develop a city to produce more than 100 culture per turn
Energize - Develop a city to produce more than 100 energy per turn
What Was Once Only Imagined - Develop a city to produce more than 100 science per turn

These are more than achieveable in the late game. Aim for bonuses that gave +% to Culture/Energy/Science. Also keep in mind that the production options to turn production in Culture/Energy/Science DO count for these achievements. You don't even have to progress the turn.

Making Way for a Hyperspace Bypass - Raze 100 Alien nests

This one can take a while. Alien Nests tend to always pop up in the same locations so when you find one, station a unit or 2 next to where it was spawned and eventually it will pop up all over again. This is most efficiently done on a Massive map with one other player on Sputnik difficulty. Just send military units out in seach of nests and camp their spawn locations.

The Only Good Bug is a Dead Bug - 500 Aliens Killed

This might take a bit of time. Alien Nests spawn aliens periodically, so if you park units by a nest, you can camp the aliens it produces over time. Keep in mind that being extra aggressive towards aliens can lower your opponents' opinions of you and can also enrage aliens in general, although a single level in Harmony mostly keeps that from happening.

Walk Without Rhythm - Killed Siege Worm

Siege Worms are an uncommon variety of alien. They typically pop up from nests that stay undisturbed for too long. They only move one tile at a time and can only travel on land, so they are easy to defeat with even a Level 1 version of the Gunner or Gunboat. You may also get this achievement if a Siege Worm decides to hang around one of your cities. They're unlikely to run very far away and will usually wander back into range without healing itself.

42 - Research All Techs

This means to complete the tech tree in a single game. This takes longer than actually winning the game, so as to not waste time, play a game with no Victory Conditions enabled or only Domination enabled and just don't try to win. The game can still end with a Time Victory before you finish, so also work on maximizing science output and not overexpanding, which increases the cost of techs.

C'mon you apes, you wanna live forever? - Fully explore the Might virtue tree
Live Long and Prosper - Fully explore the Prosperity virtue tree
Logic is the beginning of wisdom - Fully explore the Knowledge virtue tree
No bucks, No Buck Rogers - Fully explore the Industry virtue tree
A Fistful of Dollars - Unlock All Tier 1 Virtue Synergy Bonuses
For A Few Dollars More - Unlock All Tier 2 Virtue Synergy Bonuses
Once Upon A Time In Space - Unlock All Tier 3 Virtue Synergy Bonuses

You can get these achievements in separate games or all at once by fully completing the Virtue tree in the same game you complete the tech tree. Each tree has 15 Virtues to complete it. For the Tier achievements, you need 15 Tier 1 Achievements, 12 Tier 2 Achievements, and 10 Tier 3 Achievements.

Rules of Acquisition - Purchase 1000 Tiles

This is easily the most time consuming achievement. The most efficient way I can come up with is to play on a Massive Map with one AI opponent, save up tons of energy, and then plant several cities simultaneously. Once they all upgrade from outposts to cities, SAVE YOUR GAME. Buy as many tiles as you can and then reload the save. Rinse and repeat. Keep in mind that the price of tiles increases as you buy them, so saving 20,000 energy won't get you twice as many tiles as 10,000 energy.

There is no Try - Get beaten to wonders 10 times

This one is a little out of your control, but not entirely. You have to actually be building a wonder when another civ completes the same wonder. Play games against many opponents on a medium difficulty and try to go for the early wonders. It's very likely one of your AI opponents will beat you to it. Alternatively, you can outtech your opponents and have access to many wonders without building them, then when you get a notification that a wonder was completed, reload the most recent autosave and set one of your cities to build that same wonder that will soon be completed.

♥♥♥♥ Aliena - Achieve Max Level in Harmony
So Say We All - Achieve Max Level in Purity
The Sound Of Inevitability - Achieve Max Level in Supremacy

The max level for any affinity is 18. Research the appropriate techs and complete affinity quests to reach Level 18.