Here is the full list of all 36 Random Heroes: Gold Edition achievements.
Defeat the Horned Boss.
Unlock the Pirate hero.
Unlock the Ninja hero.
Unlock the Mime hero.
Unlock the Hippie hero.
Kill a Red Horned enemy.
Kill a Track Shooter enemy.
Kill a Bomber enemy.
Kill a Shredder enemy.
Kill a Hopper enemy.
Defeat the Turtle Boss.
Unlock the General hero.
Buy the SMG gun.
Kill a BrainWorm enemy.
Kill a Zombie Turtle enemy.
Kill a Worm enemy.
Defeat the Muscled Boss.
Unlock the Dino hero.
Unlock the Professor hero.
Unlock the Hobo hero.
Buy the TwinShot gun.
Unlock the Agent hero.
Unlock the Trucker hero.
Find a Crystal Skull.
Unlock the Lunatic hero.
Kill a Zombie Runner enemy.
Buy the Magnum gun.
Unlock the Gentleman hero.
Kill a Flyer enemy.
Kill a Zombie Shooter enemy.
Unlock the Cop hero.
Kill a Zombie enemy.
Unlock the Princess heroine.
Unlock the Punk hero.
Buy the Shotgun gun.
Buy the Plasma Pistol gun.