Here is the full list of all 13 Teenage Mutant Ninja Turtles: The Cowabunga Collection achievements.
Finish Teenage Mutant Ninja Turtles III: Radical Rescue (GB)
Finish Teenage Mutant Ninja Turtles (N)
Finish Teenage Mutant Ninja Turtles III: The Manhattan Project (N)
Finish Teenage Mutant Ninja Turtles II: The Arcade Game (N)
Defeat Karai in Teenage Mutant Ninja Turtles: Tournament Fighters’ Story Battle (SN)
Finish Teenage Mutant Ninja Turtles (Arcade)
Finish Teenage Mutant Ninja Turtles: Turtles in Time (Arcade)
Finish Teenage Mutant Ninja Turtles IV: Turtles in Time (SN)
Finish Teenage Mutant Ninja Turtles: The Hyperstone Heist (GEN)
Finish Teenage Mutant Ninja Turtles II: Back from the Sewers (GB)
Finish Teenage Mutant Ninja Turtles: Fall of the Foot Clan (GB)
Defeat Shredder in Teenage Mutant Ninja Turtles: Tournament Fighters’ Story Mode (N)
Defeat the real Karai in Teenage Mutant Ninja Turtles: Tournament Fighters’ 1 Player Mode (GEN)