Originally this achievement glitched on me. I deleted my save "from everywhere". And then made sure to keep track of how many kills I had.
This achievement popped when I had reached a total of 256.
Make sure to let the leaderboards register after each level (I repeated 1-4) as I think I clicked a little too fast one of the times potentially causing it not to sync and count the kills for that attempt.
If anyone experiences anything different, let me know so I know what to add to this solution.