Here is the full list of all 49 Young Justice: Legacy achievements.
Complete 1 mission co-op mode
Defeat 200 enemies
Defeat Sportmaster
Defeat Cheshire
Defeat the first 15 enemies
Obtain all Trophies
Defeat Killer Frost
Defeat Icicle Jr and Sportmaster
Defeat Bane
Defeat Black Manta
Defeat Psimon
Defeat Riddler
Defeat Blockbuster
Finish the game
Complete the game on Hero Difficulty
Obtain 50% A Grades
Obtain an A rating in all Missions
Use all the powers of the heroes
Destroy 350 objects
Activate Squad Boost 20 times
Defeat 500 enemies
Reach level 10
Reach level 20
Reach level 40
Complete 5 missions co-op mode
Complete 10 missions co-op mode
Complete all missions in co-op mode
Get half the Dioramas
Get all the Dioramas
Get all Red Arrow Journals
Collect half of the Alternate Costumes
Collect all the Alternate Costumes
Unlock all characters
Buy half of the enhancements
Buy all enhancements
Complete 1 challenge
Complete half of the challenges
Complete all challenges
Get 50% A Grades
Get all A Grades
Answer 5 YJT question correctly in a row
Reach a Score of 500 in the YJT
Reach a Score of 1000 in the YJT
Reach a Score of 5000 in the YJT
Don’t break any Statue in the Museum Stage during “A Hostage Crisis”
Escape the Avalanche Stage during “Go with the Flow” in less than 5 minutes
Don’t be damaged by any trap in Atabey’s Temple Stage during “The Godess Shrine”
Avoid search light in Gotham Docks
Don’t activate the Statues in Esagila’s Temple in The Warriors Stage during “Tag Team”