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