I can't speak specifically, but I would direct your attention to Justinian's Corpus Juris Civilis, with particular attention to the Digest within. That will give you a lot to look at regarding violent crimes and other matters of criminal and civil law.

That being said, while you'd have to go to archives to find out specific statistics regarding violent crime in specific spans of time, Constantinople wasn't much different than any other Medieval Mediterranean city; the Nika Riots in 532 C.E. show that even when there was tight civil imperial leadership things could still get out of hand.

After Justinian, things fluctuated wildly between excellent civil administration and downright anarchy, so it's difficult to accurately characterize justice in application for Medieval Constantinople.

You might find some helpful information in Lars Brownworth's book on the Byzantine Empire. It's a fun and easy read, and you can get a really good sense of civil society without too much scholarly overhead.