Here is the full list of all 42 Monolith achievements.
Become the reigning tag champion.
Reach the Excavation (Floor 1).
Reach the Archives (Floor 2).
Reach the Maintenance System (Floor 3).
Reach the Bellows (Floor 4).
Reach the Sanctum (Floor 5).
Defeated the Overlord.
Get an S Rank in Time (Under 20 minutes).
Get an S Rank in Damage (10 hits or fewer)
Get an S Rank in Exploration (Visit all rooms)
Get an S Rank in Secrets (Find all secrets)
Get all 4 S Ranks in one run.
Finished a run with a score of 100,000.
This is what you wanted, isn't it?
Won without getting any upgrades.
Won without getting any weapons.
Leave that poor cat alone.
Pass the seal. (Floor 6)
Tarnish notte the majesty! (Acquired all Hats)
Finished a run with a score of 200,000.
Won the game on Intense Lethality.
Won the game on Sudden Death Lethality
Protect Blastcore's smile.
Reach the Temple.
Reach a highscore of 20 minigames in the Hub.
Take a screenshot containing 100 bullets.
Admire a fine portrait.
Finished a run with a score of 300,000.