Here is the full list of all 43 Last Kids on Earth and the Staff of Doom achievements.
Revived a friend in multiplayer
Reached level 5
Reached level 10
Reached level 15
Reached level 20
Completed all main and secondary quests in District 1
Completed all main and secondary quests in District 2
Completed all main and secondary quests in District 3
Completed all main and secondary quests in District 4
Defeated Vine Claw
Defeated Scrapken
Defeated Spikipede
Defeated King Wretch
Defeated the Slime Impostors
Defeated Meathook
Defeated Hairy Eyeball
Defeated Neon Bosses
Crafted a tree house turret
Crafted all tree house turrets
Fully upgraded all tree house turrets
Crafted a weapon
Crafted all weapons
Fully upgraded all weapons
Crafted an armor
Craft all armors
Fully upgrade all armors
Upgrade BigMama to level 2
Upgraded BigMama to maximum level
Upgraded tree house to level 2
Upgraded tree house to maximum level
Inflicted 5000 damage points with Rover
Had Skaelka perform 5000 spins
Had Bardle heal 5000 health points
Had Biggun absorb 5000 damage points
Stunned 250 enemies using Jack's BooMerang
Defeated 250 enemies using June's Toxic Frisbee
Used Dirk's Sticky Grenade against 250 enemies
Distracted 250 enemies using Quint's Cymbal Monkey
Completed the Bestiary
Completed a Big Mama challenge
Completed all Big Mama challenges
Found one comic book cover
Found all comic book covers