Here is the full list of all 32 DC's Justice League: Cosmic Chaos achievements.
Reach level 30
Complete all of Lobo's Bounties
Drive the forklift circuit
Ride a ride at the Boardwalk
Complete Blue Snowman's mission line
Complete Poison Ivy's mission line
Conquer Starro the Conquerer
Arrive in the 5th Dimension
Complete Bizarro's mission line
Rescue The Flash
Ride the floatie
Unlock every P.O.R.T.
Fully upgrade a Support Hero
Complete Bethany Lee's mission line
Rescue Aquaman
Complete Ares' mission line
Fully upgrade an Artifact of Justice
Thwart Operation Snootilyboopikus
Fully upgrade a Heroic Ability
Complete Clayface's mission line
Rescue Green Lantern
Upgrade a Support Hero
Defeat 6 enemies with a single attack
Ride a skateboard as Superman
Rescue Snapper Carr
Upgrade an Artifact of Justice
Equip an Artifact of Justice
Lasso a fish into water
Defeat 3 enemies with a single attack
Get Mount Justice Upgraded
Get your first costume from Booster Gold
Upgrade a Heroic Ability