Here's the list of achievements in Abyss Odyssey.

For obvious reasons the guide isn't spoiler-free.
For any note, detailing and corrections, please let me know.
- This current guide is updated to the Halloween Warlock mask: the game is updated frequently by the developers and they add achievements very often, so be patient if you don't see still one listed, just let me know and I'll try to solve it for you ^^ -

• ONCE UPON A TIME...
  First one and automatically given when ending the prologue.
  
  
• THIEF
  In the abyss there're occasional spots for merchants, these misterious characters sell you various goods for an high price.
  Just under the list of goods you'll notice the option for fighting the merchant: if you choose to do so and kill him you'll be awarded with this.
  
  Note: he can be captured like any other creature, for capturing a creature the character has to be of higher level than the creature's one.
  
  
• DANCE WITH THE DEVIL
  When looking on the map, you'll notice a violin symbol on some of the levels.
  Explore the map until you reach that level, then follow the violin sound to find this demonic version of Paganini.
  Accept his offer and you'll get a special item or a captured enemy.
  He will disappear to be found, later, in a random level and hostile to you.
  Kill him to get this achievement.
  
  Note: he can be captured, granting you the DREAM HUNTER too for his high level over 30, and is quite strong and agile with a third mid-air jump ability.
  
  
• ASSASSIN OF KINGS
  When looking on the map, you'll notice a devilike symbol on a level
  Explore the map until you reach that level and be prepared to fight a mid-game boss that is quite strong.
  
  Note: he can be captured, granting you the DREAM MASTER too.
  
  
• LION TAMER
  This is unlocked after beating a boss that randomly spawns out of one of the 3 entrances to the Abyss.
  The conditions for its activation are: at least you reached the cemetery-type levels, near the bottom of the Abyss, before dying, or a 1/3 chance to confront it after beating the warlock for 3 times.
  This event comes with a cutscene, and may occur in any of the 3 entrances.
  Anyway I'll suggest you to choose the mid-one entrance: I have done many runs and confronted the "Living Creature" only 3 times and always in Pocuro Park.
  So keep trying if it doesn't shows up.
  Be prepared for a tough fight.
  
  Note: it can be captured, and its level is 45. By the time you can capture it probably you'll have already the 3 DREAM achievement, see below.
  
  
• LORD OF SLEEP
  Just reaching the end of the map and killing the head of the nightmares will get this one.
  
  
• WISHING WELL
  When looking on the map, you'll notice a yellow colored level.
  There're a lot of treasures inside as well a npc and weak butterlies creatures.
  In the center there's a big golden fountain, standing near its base (not so far from where the npc is) you'll will get the option to interact and throw money in it.
  The total money needed is around to 25000-30000.
  Upon reaching this target, Pincoya will appear and she will be available as character.
  It's possible to do donations in subsequent different runs.
  
  
• DREAM CATCHER
  Easily done with the capturing spell on a Peacock fighter or any lvl 10 or more creature.
  
  
• DREAM HUNTER
  You need to capture a lvl 25 or more creature.
  I prefer Paganini for that, you'll get later the achievement but 2 with one try.
  
  
• DREAM MASTER
  You need to capture a lvl 40 or more creature.
  The target for this one is Verbum Dei.
  
  
• PERSISTENT ADVENTURER
  While playing the game occasionally creatures drop the Warlock's diary pages.
  Some are dropped by mid boss and require more than one try to make them drop.
  When you'll have enough entries in the diary, this one is unlocked.
  
  
• ULTIMATE KATRIEN
  The first of 3 similar achievement: it's obtained by gaining skill points and assigning those points to fill all the slots of one possible combination of the 3 moves selected in the moves customization available at the monuments or other special locations. This's for Katrien.
  
  
• ULTIMATE MONK
  It's obtained by gaining skill points and assigning those points to fill all the slots of one possible combination of the 3 moves selected in the moves customization available at the monuments or other special locations.This's for Ghost Monk.
  
  
• ULTIMATE PINCOYA (last of the original 14 achievements)
  It's obtained by gaining skill points and assigning those points to fill all the slots of one possible combination of the 3 moves selected in the moves customization available at the monuments or other special locations.This's for the Pincoya.
  
  
• GRANDMASTER
  It unlocks automatically upon completing the game on the new feature nightmare difficult.
  The Friendly Fire option must be active, too ( so set it to "yes").
  Even if you're playing solo.
  
  
• THE GOLEM
  There's a new mid-boss from another game of ACE developers (Zeno Clash).
  You have to find him looking on the map: check the map before you begin your descent into Abyss, look for a reddish evil face icon, if you see 2 types then the one with longer horns costantly moving is Verbum Dei, the other one is Golem.
  -At the moment I'm writing this, the achievement is broken and got it for buying this enemy soul from the priest in the cathedral: even if you beat golem you'll not be awarded with the achievement, besides it's already in the list of monsters souls in the shop and it seems you can't capture it even if you're at max lvl. - hope they fix it soon -
  
  
• ONLINE CHAMPION
  Win 5 online matches during the same game session.

Gaming tips

As mentioned in the comments here're some simple tips that may help getting to broke the nightmare.

Money carries over runs, so if you die a lot try saving the money at least. -completely different compared to real life huh? :D

While the items the merchants in the Abyss are tempting, sometimes, you should avoid completely any shop from those guys.
Items are often fairly overpriced.
Only when you find health potions, those are a must buy.
Instead of shopping, consider to kill them for the experience: they give a lot of exp. and if you're about to lvl up remember that a percentage of health is recovered doing so.
But remember that they're still a challenge, don't go rushing for it.

Give always the gold to the Jackal minstrel, it's a bit of exp. and it's useful for the same reason mentioned above.

After a successful run, or if you're already hoarding money, consider as new starting point Pocuro park or the Cathedral because you may respectively buy Camp badges or creature souls from npc at an average to low price.
The npc are a Firefighter in Pocuro near the tent - Camp Badges
a priest in the Cathedral - Creature souls (you have to jump on the scaffold to reach him)

Build your special moves with at least one move with only skill points assigned to Mana %.
Reasonable choice for it is a move that does multiple hits.
The faster you get the mana filled is for the best:
you'll get the creature's souls, thus being able to transform and a sort of second life/chance
the magic attack is quite powerful, pauses action and AoE.

After you use the magic attack, start again building up mana even if you have already a creature:
you may found soon another creature more powerful than you're actual one, so you could consider swapping it.
Another reason to swap may be for having a fresh, full health creature.

If you spot a Monument altar but you don't go near it,
you're low on health,
and you have only a magic weapon and no items/souls in inventory:
don't miss the chance of being resurrected by your nice and faithful guard-follower that swaps with you when your hp reach zero, grab your weapon and backtrack to the monument to come back at full health with your weapon.
Do so intentionally.

Thanks for reading and enjoy the game :)