The third tier of War of the Gems requires you to activate infinity gems 30 times.
In arcade mode, you start with 1 gem. You can knock gems out of your opponent with special moves/infinity counters and pick up additional gems. You'll start with more gems in each match as you progress further into arcade mode. You can activate as many gems as you can carry per match.
Cycle through and choose which gem you want to use by pressing the button. They're activated by inputting +. The screen will pause for a moment and show which gem you've activated. Do this whenever possible as you play through arcade mode as gems can give you an advantage during a fight and you can earn progress towards this achievement at the same time.
Gems are a little different in multiplayer. You start with no gems, but gems will enter the field when someone scores a "first attack" and when a players health is low enough to trigger the "danger" notification. You can still knock gems out of your opponent using specials and infinity counters as well.