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