Here is the full list of all 44 Nuclear Throne achievements. It takes around 200 hours to unlock all of the achievements on Windows.
Loop as every character.
Reach 4-? as Crystal.
Reach the Nuclear Throne in under 10 minutes as Plant.
Unlock Hard Mode.
Defeat the Technomancer as Steroids.
Defeat Mom as Rebel.
Reach the Nuclear Throne with Melting without choosing Rhino Skin or Strong Spirit.
Defeat Hyper Crystal as Horror.
Defeat the Technomancer.
Defeat Hyper Crystal.
Defeat Mom.
Unlock a Golden Weapon for every character.
Reach Level Ultra as any character.
Reach 7-3 on a Daily.
Loop past the Nuclear Throne.
Defeat Throne II.
Unlock a Crown as any character.
Loop the game.
Reach 5-?.
Defeat Horror.
We hereby sincerely apologize.
Eat a Hyper Weapon as Robot.
Reach the Nuclear Throne.
Defeat the Nuclear Throne.
Reach 6-1.
Defeat Lil Hunter.
Reach 5-1.
Reach 2-? as Eyes.
Defeat Big Dog.
Unlock a Golden Weapon as any character.
Reach 3-?.
Reach 3-1.
Reach 2-1.
Die.
Defeat Big Bandit.
Reach 2-1 on Hard Mode as Chicken.
Defeat Captain as Rogue.
Unlock a Golden Disc Gun or Golden Nuke Launcher.
Reach Level Ultra as Skeleton.
Play as Frog.
Sit on the Nuclear Throne as headless Chicken.
Unlock all Crowns as one character.
Defeat Captain.
Get 100% of the unlocks.