Update: This challenge has now finished and can no longer be earned.
The most efficient way to get this is through the Episode 1 - Buck National vs. The Apocalypse challenge. You unlock it fairly early on in the campaign and it is in fact part of the story as well. The challenge is located in the north area of the left island.
The challenge repeats itself over 3 minutes where it asks you to get bounce kills, grind kills, Blower kills and train kills. Simply rinse and repeat.
You can use other weapons to get through the challenge cycle more quickly but don't forget to switch to the Flaming Compensator for the Blowers :)
As Reborn Insanity
notes, be sure to snap the TA app with this solution open before hitting 200 so you'll unlock: