A lot of the achievements are self-explanatory and a lot are also story related, so you cannot miss them. I'll only talk about the ones that might require some explaining.

About Medal achievements (Wood / Bronze / Silver / Gold)

The game has four different difficulty settings, each one gives you a different type of medal.
If you play on higher difficulty settings, you will also get the lower difficulty medals on the same run. (I completed the game on Easy, then went right to Extreme, unlocking Bronze, Silver and Gold medals at the same time).

About Multiplier Achievements

In this game, you increase your score multiplier by filling up the Multiplier Bar, located at the top right of your screen. To do this, you must trigger the traps you find across the level, while being careful and not getting killed. Different traps fill the bar at different rates, and you can trigger the same trap more than once to increase your multiplier without moving much.
To get the high multiplier chest / gem achievements, you'll have to play
on the higher difficulties (Hard / Extreme), so more traps will spawn over the same space.

Hangman / Super Hangman:
You can get Hangman with Ann, but Super Hangman can only be achieved with Dan, since his arms are stronger. both achievements unlock when you touch the ground.

Narrow Escape / Escape Artist / Master Escape Artist:

These achievements are a bit confusing since there's no ingame counter of how many traps you've activated without dying. And for me, all of them unlocked when I inevitably got killed by a trap (was it trap 101? trap 150? who knows).
Also, since I couldn't determine if the same trap counts more than once, I had to finish one of the harder levels (one in Asia) on Medium difficulty without dying. Just trigger as many cracked blocks as you can, and stay away from fire and water traps (too quick for escaping).

Preemptive Strike:

This is done with Ann's pickaxe on a spike trap that's turned sideways. Drop down from a higher point, and you'll see the percentage number appear. Use your pickaxe and avoid getting killed.

Almost There:

You will get this by taking the item at the bottom of a temple, then dying by either a trap, a normal block pushing or crushing you, or the floor falling out if you wait long enough.

Specific Map Achievements

Break No Sweat:
"Amsterdam: Finish the level without pushing or using your pickaxe."

To get this, you must finish the Amsterdam level without using the characters' special abilities. No pushing, no pickaxe allowed. Just play it on Easy and wait for the blocks to disappear naturally. I'd choose Dan over Ann here, cause if you push the special button by mistake Ann will use her pickaxe instantly, while Dan also needs a direction key to be pressed.

Die Already:
"Naples with Ann: Survive three minutes with the map and get out alive."

When you get to the bottom of the Naples ruins, check the door where you want to exit and see if it has a reachable block directly above it. Quickly take the map, thus triggering the self destruct sequence. Get up onto the block above the door, and stay still while the onscreen counter goes down. When the timer ends, jump down and mash the Use key and you will go through the map exit while in mid-air (the floor will be gone by then).
Do NOT hang on the edge of the block, since Ann's strength runs out way faster than Dan's.
If you do this on the first try, you will also unlock Lucky Survivor (one minute) and Skilled Survivor (two minutes).

Die Hard:
"Resistencia: Finish the level without dying."
Light Footed:
"Zaheda: Trigger no traps and don't die."
On both Resistencia and Zaheda, you have a "do not die" achievement. If you die before reaching and opening a door, it means the achievement is unobtainable from that point on: you must exit to the diary menu and restart.

I reccommend you choose Dan for both levels, since he can protect himself from falling blocks both normal and cracked(the trap blocks), he can push things off the tower if you're on a corner, and he can double jump to escape tight spots. In Zaheda, you'll have to be VERY patient and leave all traps to disappear by themselves. Stay away from cracked blocks, sometimes they will trigger from where you cannot see them when the solid block that was previously blocking them disappears. Also, keep in misnd that destroying a cracked block that's locked by another block on the other side does NOT count as triggering.

No Shortcuts:
"Rasht: Finish the level on medium without using any doors."

This is pretty similar to Die Hard and Light Footed achievements, only with the difference that you can die and keep trying without restarting the map. Just don't pick up any keys, or use them exclusively on chests. i did this one with Dan for all the reasons mentioned above, but since it's only Medium you can probably finish it with Ann too.

What Treasure?:
"Stuttgart: Finish the level without collecting any treasure."

Choose Ann, move quickly and stay away from the wall openings that spit out coins and gems. I don't think chests are available in here. Use the door savepoints, and if you catch treasue by mistake you can kill yourself and resume from the last savepoint.

Speedrunner:
"Paris: Finish the level within 40 seconds without using any doors."
This used to be the most difficult achievement of the game, when it took place in Puerto Montt. But the devs moved it to Paris, and now you only have to set the difficulty to Easy, choose Ann, and mash that pickaxe button. Well, maybe learning the layout could help a little bit.