Here is the full list of all 24 Nickelodeon All-Star Brawl achievements. It takes around 10-12 hours to unlock all of the achievements on Windows.
Clear Arcade with all (standard) brawlers
Clear Arcade with 15 brawlers
Play 50 local battles
In Sports, win a game with basketball vs CPUs with a 10 score lead
In Sports, win a game with medic ball vs CPUs with a 10 score lead
In Sports, win a game with dodgeball vs CPUs with a 10 score lead
In Sports, win a game with soccer ball vs CPUs with a 10 score lead
In Sports, win a game with rugby ball vs CPUs with a 10 score lead
Play at least once with all standard brawlers in local battle
Clear Arcade with 10 brawlers
Complete 50 online quickplay matches
Defeat two level 9 NPCs in a team-battle where you are alone on your team
Defeat three level 9 NPCs in a team-battle
Clear Arcade with 5 brawlers
Complete an Arcade run without losing a single stock
Compete in 10 local battles
Defeat an opponent with a higher rating than you 5 times in online mode
Complete an Arcade run on the highest difficulty
Defeat a level 9 NPC in a 1v1 battle
Complete 10 online quickplay matches
Clear Arcade with 1 brawler
Complete an online quickplay match
Defeat an opponent without losing a single stock in an online battle
Compete in your first local battle