Here is the full list of all 36 Random Heroes: Gold Edition achievements worth 1,000 gamerscore.
Defeat the Horned Boss.
Unlock the Punk hero.
Unlock the Princess heroine.
Unlock the Pirate hero.
Buy the Plasma Pistol gun.
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.
Unlock the Ninja hero.
Buy the Shotgun gun.
Unlock the Mime hero.
Unlock the Hippie hero.
Unlock the General hero.
Unlock the Dino hero.
Buy the SMG gun.
Kill a BrainWorm enemy.
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.
Unlock the Agent hero.
Unlock the Trucker hero.
Find a Crystal Skull.
Kill a Zombie Runner enemy.
Unlock the Lunatic hero.
Unlock the Gentleman hero.
Buy the Magnum gun.
Kill a Flyer enemy.
Kill a Zombie Shooter enemy.
Unlock the Cop hero.
Kill a Zombie enemy.