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