Teenage Mutant Ninja Turtles: The Cowabunga Collection

Pick "Story Battle" with default settings. You DO NOT have to play on 7, default is 3 and I unlocked the achievement.
Pick Leo. He has a move that helps a lot. It's a projectile. You can't solely spam it, but you can use it a lot. Use to rewind if you get hit. I'm not very good at fighting games, but it seems to me like you do more damage if you hit them right before their move hits you. There's around 8 (maybe 9) people to fight, 2 rounds each, 60 second rounds. If you have more health than them when time is up, you win the round.
Basically, abuse rewind to chip away at their health without taking damage yourself. Try to kill time by jumping away.
Most importantly, SAVE at the start of every match. That way if you get yourself into a situation that you can't rewind out of, you won't have to start over.
*Edit: SiegfriedX pointed out that you can change the time to 30 seconds. My wife did this successfully as well. I'm not sure about other settings or enhancements though

You can't use the watch mode exploit to get this achievement, its not played on the correct mode for the achievement.

Here's what I did for this achievement. Its pretty easy:

From the game select screen (before loading up the actual game): go to the SNES Teenage Mutant Ninja Turtles Tournament Fighters and press . Set Ultimate-Attacks In Story Mode to ON. Its not necessary, but it can speed things up.

In-game: you do not need to adjust any options or settings unless you want. Game level can be kept at the default (3) and you will still get the achievement.

I'm convinced there's confusion over ending vs achievement. You will fight Karai on this difficulty level and get the achievement once she's beaten, but you won't get an ending. Way back in the NES/SNES days, game rentals were big business and cut into publishers' game sales pretty badly. Konami's genius solution to get back those lost game sales was to deny players an ending until they could beat the game on hard. Yes, that's right! Konami honestly thought this would discourage game rentals and get people to run out and buy their $50-$60 SNES games since no one could possibly ever be able to beat the hard difficulty across one or more $5 2 to 3 day rental period. Right? This is why almost every SNES Konami title offers a lame text ending telling you to play the game on a harder difficulty if played on easy/normal!

Anyway...

On the title screen: select Story Battle

Pick Raphael

You can win most battles using the Power Drill move (charge , or )

The computer will block it most times. This is what you want. As the move ends, press and hold towards or away from your opponent and hammer the or buttons. About 90% of the time, you will be standing right next to the opponent and will throw them as soon as power drill ends.

Sometimes the computer will be cheeky and throw you instead.

If the throw fails, or the CPU throws you instead, remember you can rewind time and retry as many times as you like by holding down

If you enabled Ultimate-Attacks, the power drill move will charge up the bar pretty quickly,

Once charged, it can be activated pressing + at the same time.
If this move connects, blocking or no, it will usually destroy the CPU.

Use Donatello, Use Y and X moves, Down Corner Forward does the ground attack, push away on the d-pad to block.

See the video for a better understanding.

Set Game Level 7.
Game Time 30.
Game speed Normal.
Select Story Battle