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