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.
Complete an Arcade run on the highest difficulty
Clear Arcade with all (standard) brawlers
Clear Arcade with 15 brawlers
Clear Arcade with 10 brawlers
Clear Arcade with 5 brawlers
Complete an Arcade run without losing a single stock
Play 50 local battles
Defeat an opponent with a higher rating than you 5 times in online mode
Complete an online quickplay match
Complete 10 online quickplay matches
Complete 50 online quickplay matches
Defeat an opponent without losing a single stock in an online battle
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
Compete in 10 local battles
Defeat a level 9 NPC in a 1v1 battle
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 1 brawler
Compete in your first local battle