Without asking the kid, this reeks of an inside joke or internet meme. My hunch says that the reference is to Grizzly Adams, Bear Grylls, or the animal. (The first two seem unlikely since the kid probably has no idea who Grizzly Adams is and Grylls is distinctly different from Grizzly.)
If the reference is to the animal, than achieving grizzly status would mean to become a difficult to kill beast that is able to tear through enemy forces. Within the context of of Call of Duty video games, this would mean something similar to going on a rampage or killstreak. Notches above "grizzly" could be unstoppable, godly, demigod, etc.
Edit: Truncating to "grizz" makes sense, actually, and refers to a style of play that involves using sniper rifles without a scope and trying to spin around before shooting someone. As the comments below mention, it is named for a famous FPS player: zzirGrizz. YouTubing the name should reveal a few dozen clips of examples.