This achievement appears to be either buggy or have an inaccurate description (possibly both). Ideally, though, it's pretty straightforward.
What you're supposed to do it go to one of the "Battle Arena" areas (accessible from either the fast travel map or in the hub areas, often right after you finish a story mission). Head in here and play all three game modes once (you have to go down to "customize game" to change the mode).
There are three modes: "Samurai Showdown" (where you fight over four flags, and the longer you hold on to them, the more points you get), "Mystic Bounty" (where you collect orbs for points), and "Ultimate Ultimate Weapon" (where all players fight for a box and hold on to it to score points).
While the description says "play" each mode, it appears you have to actually win at least some of your matches. If you play all three and the achievement pops, great. If it doesn't, try re-playing a mode that you didn't win in the first time, and try to win (in my case, I re-played Samurai Showdown and won, but never actually won a game of Ultimate Ultimate Weapon). If you're having trouble, put in some dummy controllers so you have fewer AI to fight against.
It's entirely possible you won't have to deal with this issue at all, but if you do, hopefully this trick will help. If I made an error or you have anything to add, leave a comment and I'll edit (don't just down-vote with no comment, that doesn't help anybody).