You need to get 200 ring captures and 100 ring breaks in a public match.
This can be done in king of the hill, escalation. At the time of launch for this patch it can also be done in Blitz.
Apart from boosting it with a full lobby (you'll need 6 xboxes and 10 controllers), your best options are:
For break play blitz while you can. Breaks do NOT show up on scoreboard in match, but they DO count towards achievement.
Instant captures make this very fast. Even without cooperation from anyone on either team it isn’t hard to get 10-15 per round. Best way is to run/roll through ring and out the other side, get in cover/heal then do it again. You can sometimes get four or five breaks in a row depending on how effectively they are defending it. Worst case you can usually get one suicide break, then respawn and repeat. You can’t farm ring captures by capturing it and leaving it though, I’m guessing that an enemy needs to capture in between.
For captures coop vs AI is probably your best bet but will work better if your team is cooperating with you. Capture the ring then leave and back up a bit so the AI will break it, then recapture.
If you do manage to get a boost going, blitz would be your best bet, failing that KoTH. It would probably take you all of half an hour once you get together. If you don’t know how to ring boost, refer to the solutions for gears 2. Party like it’s 2008!Solution for Veteran Gear in Gears of War 2