This will likely come from trying to complete all of the other achievements, but I think there's a pretty easy way to do this and the inspiration comes from a different achievement:
In this guide, the author details using the EX Options. Here's what you'll do:
1) Access the EX Options Menu under Collections Menu
2) Turn On Virtual Player Count up to 4 Players (it costs Credits to unlock)
3) Select the Character Lemon
4) Go to Zone 2, Level 1 (Hive)
5) Kill until you either lose or hit the achievement
The author also mentions the Mega Weapon Ex Option, but I didn't use it. You'll easily rake in 1000+ kills every run of the level. 15k kills will pop in no time from there.