Super straightforward. Doesn't matter the difficulty you play on or what character you are.
If you get a dedicated online session going you should come pretty close to racking up 200 KOs through the course of the entire campaign. If you come up short or can't seem to get a game to play through the whole story, head to Stage 3 - Leo's Place while playing in online mode.. This level is the easiest spot to rack up the most enemies in the shortest time. Give that a go and you should have the achievement unlocked in no time!
It should also be noted that you'll want to make sure to finish the levels to make sure your KOs are being counted. Online doesn't seem to save your stats as often and you may lose progress towards online achievements when you quit before the end of a level.