This trophy will more than likely come through natural progression of Story mode.
There are plenty of weapon pick-ups on every single level; simply pick them up and hit square to use them. You can easily rack up several kills with the large boxing glove and rocket weapons. And of course, there are several battle arenas throughout Story mode.
If you somehow manage to complete all 71 levels and don't have this trophy, then selecting a battle arena from The Space Bass planet is the easiest way to grind the 50 kills, as most (if not all) of your opponents will be named Hoard racers.