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