Here is the full list of all 44 Nuclear Throne achievements. It takes around 200 hours to unlock all of the achievements on Windows.
Eat a Hyper Weapon as Robot.
Reach 2-? as Eyes.
Reach the Nuclear Throne.
Unlock a Crown as any character.
Defeat the Nuclear Throne.
Reach 6-1.
Defeat Horror.
We hereby sincerely apologize.
Defeat Lil Hunter.
Reach 5-?.
Reach 5-1.
Defeat Big Dog.
Unlock a Golden Weapon as any character.
Reach 3-?.
Reach 3-1.
Reach 2-1.
Defeat Big Bandit.
Die.
Loop past 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 the Nuclear Throne in under 10 minutes as Plant.
Unlock a Golden Weapon for every character.
Defeat the Technomancer as Steroids.
Reach 2-1 on Hard Mode as Chicken.
Defeat Mom as Rebel.
Defeat Hyper Crystal as Horror.
Defeat Captain as Rogue.
Reach Level Ultra as any character.
Unlock a Golden Disc Gun or Golden Nuke Launcher.
Reach 7-3 on a Daily.
Reach Level Ultra as Skeleton.
Play as Frog.
Sit on the Nuclear Throne as headless Chicken.
Defeat Throne II.
Defeat Mom.
Defeat Hyper Crystal.
Defeat the Technomancer.
Unlock all Crowns as one character.
Unlock Hard Mode.
Loop the game.
Defeat Captain.
Get 100% of the unlocks.