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