Turns out boosting this achievement is actually pretty easy. If you have a boosting partner, try to search at the same time and 75% of the time you should match up together. You can take turns quitting out and it gives the other person the win and an opportunity to pull the ring.
There's no penalty for quitting out other than losing a contract so try to put the worst qualifying cards in your lineup so you don't waste any contracts on good players.
The only way to win a season ring is to win MyTeam Limited games. This is an online mode that restricts your lineups and can only be played on Friday, Saturday, or Sunday.
After a win, you are allowed to choose from three different face-down cards as rewards. These include MT, contract packs, shoe packs, and the all-important Season Ring. After your first option, you are allowed to either keep what you've turned over or select one more card, but if you do, you must take the second card no matter what it is.
If you don't see a Season Ring on your first card turn, always elect to select another card for a chance at getting one. The three rewards offered will not always contain a ring. In fact, most of them will not. It's entirely random when a ring shows up, so you'll have to keep on winning games against online opponents in order to have a chance at drawing the ring.
I've seen reports of a season ring showing up after just a few games and some people have reported winning 40+ games without having a ring be an option to select from, so this achievement will not only require you to keep beating human opponents, but also win the game of chance that is choosing the correct reward card.