Streets of RAGE indeed! This requires no small amount of patience.
You are required to get a high score of 800,000 on Hardcore (Mania) with one credit, as your score will be reset to zero if you use a continue.
You can do it in co-op but I find it much easier to do solo so you can make ample use of the save state feature. Just save often and reload if you take too much damage. There are some tricky spots where you cannot avoid losing a life or two, especially when facing multiple ninjas or jetpack flyers, but if you persevere you will eventually pop the trophy before the end of the game.
I popped mine at the end of stage 7 on my last life using Max the wrestler.
Alternate glitched method:
If the first method has you raging hard, try this (courtesy of BakaSensei2011):
Note: Each press of the attack button while mid-air with the boss will earn you points (works well with rapid fire turned on). But as soon as you land, let go of attack or you will have to start over.