This is achieved in the first level/area of the game - after you kill a bunch of grub looking creatures hanging from the ceiling (they're like a white cylinder shape, you'll know them when you see them), Rocket will challenge you to see who can kill the most, complete with a retro pixel scoreboard at the top of the screen.
Going forward into the level, you'll find more of these hanging grub piles - Rocket is pretty quick at killing them, so try and shoot them out before he gets a chance. You'll probably fall behind early on, but don't worry, there's plenty of areas to catch up.
One really good section to boost your score is when you head through the eye of the giant robot face. You'll see at least 10 of the grub piles in this area and if you stand at the top before dropping down, you can shoot a lot of them out from a distance really easily. Rocket will drop down and not do a lot here, giving you an extra advantage. Make sure to check all around this large circular room - some are hiding down below and behind the pink goo that covers the level, but they're not that hard to find.
Another good section is when you have to enter through the engine of a ship (just after sending Rocket into a vent to open a large door) - there are quite a few piles as you move through this section and Rocket doesn't seem to really go for many of them.
Once you've racked up your score here, just proceed through the level until you enter a dark room and have to kill a final grub pile to progress the story. Here, if you're ahead of Rocket, you'll get the achievement. My score was about 31-18, so I was fairly comfortably ahead without knowing where any of them were in advance - you shouldn't have any problems beating him.