You only need to defeat him to pop the achievement. As soon the cutscene ends when you find him (the second boss; you'll find him as the glowing mutagen canister on the map), get on his right side and start spamming your attack and the forward shellkicker. Keep doing this and you'll continue to smash him into the crates on the left. Eventually he'll start flying above you and release some mousers. Do your best to hit him with shurikens and jumping attacks until he dies or returns to the ground (spam attacks again until dead).