Start up beast with the following settings:
Map: Checkout (you want a small one)
You can get this while going for:
Wild Ticker: This will be your hardest kill to get, and so it's best done on wave 1. I started off as the Savage Drone and killed everyone except the last guy. I proceed to let this guy take my health down almost to the end and then downed him (don't fully kill him) then go kill yourself on a turret or a spike fence. respawn as a Wild Ticker and just walk up to the guy and kill him. It'll take some practice but this is the easiest way to do this alone.
*Also if you down a guy and die, then respawn as a wild ticker and then the guy dies (without you atacking it) it still counts for the wild tickers kill.* - Found by quinnpm
Ticker. Just keep at it. The ticker is easy to go around a corner and kill a guy. Make sure you use your run (hold A) feature. If your having trouble use the same strategy as I describe for Wild Ticker.
Wretch: make sure you utilize that the wretch can climb stuff (A) and leap through the barriers (RB). I've found that this is best on Wave 1 as well. You can use the same strategy as for the Wild Ticker, but I've found that this is doable with just normal play.
A list of each type if you want to cross them off:
You can also keep track of the ones that you need in the Stats & Awards / Achievements / Wreaking Locust Vengence. But this has to be done from the main menu.
*Note: some people are having a problem with the ticker and it not counting after playing on the same level. The solution people are having is just try on another map and the achievement will pop*