This one is a little tricky. First set the game to easy, and play the game normally, you need to beat the first stage completely (3 different screens if I remember right) There's a sub boss with a gun, and a real boss with multiple life bars.
Then in the second stage at the end of the first screen you'll fight a kangaroo (Roo) and a trainer (Bruce). Kill Bruce with out killing Roo and you'll unlock this trophy and Roo as a playable character.
Note: Unlike Streets of Rage 1 you can use your super any time and the bar will refill over time, so feel free to use and abuse that as much as you want.