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