Ultimately up to the GM, but in general, yeah, addiction to violence/adrenaline-rush can work for a Shadowrun Addiction Negative Quality. I would work from the rules backward in order to describe how it works but generally follow the "Insubstantial Substances" section as a guide. The GM and player can figure out the addiction rating and threshold for that particular character (with minimum around the same ratings as for alcohol). A dose/hour of activity could be something along the lines of starting bar fights, joining underground fighting clubs, illegal street racing, etc. (I personally would limit it to one activity for the character; fighting, dangerous driving, etc.). But I'd largely place that in the background unless interfering with or occurring during a run. "Overdosing" would mean they ended up taking that much damage after satisfying a cycle's cravings.
Then on a run, engaging in that kind of activity would open up the addiction window (11 - rating weeks) that risks them becoming more addicted. Hitting burnout stage means more lost fights (losing badly) or car wrecks or "fans"/bookies/etc. getting physical, and so on.
If it's just one GM and one player, then all of that can be role played out for the fun/terror of it. If there's more players then I'd treat it more like substance addiction and keep it in the background save for lasting effects like damage from Overdosing and what not.