To bypass fighting everybody and get to the boss trio without losing continues start a 1P game. When the screen changes to the space map have controller 2 enter the game for a 2P game. Have player 2 choose a character and select that person's home planet.
Leonardo - Bio Planet
Michaelangelo - Mirage Planet
Donatello - Futuristic City Planet
Rapheal - Magma Ocean Planet
April O'Neil - Spaceship
Casey Jones - Ice Planet
Ray Fillet - Ocean Planet
Sisyphus - Jungle Planet
You can beat up a defenseless second player and the game will give you credit for defeating that character in the tournament lineup. After you beat all eight defenseless fighters on their home stages the game will begin with the boss matches and you'll have kept all your continues.