Paraphrasing from my ODB:

There was no office of public prosecutor, but the authorities could initiate criminal proceedings. However they were often initiated by a private party applying to the authorities regarding some crime. Who that could be depended on the nature of the crime. The difference between civil and criminal procedure was "blurred", though interestingly an unsuccessful accuser was threatened with the same punishment that the accused would have suffered if convicted.

Murder was punishable by death if you were a commoner or by banishment and property confiscation if you were a notable. Death sentences may be commuted to hard labor in the mines (I don't know if that was for life) or corporeal punishment for mitigation circumstances (I don't know what those were!). Murderers typically couldn't escape execution by seeking asylum in a church, though for a period from the 10th to 12th centuries they could. When they could seek asylum, their property was confiscated and awarded to their and the victim's families. Murderers could escape prosecution if they paid a penalty to the victim's family.

Robbery was actually considered a delict unless it was particularly violent. Rape of unmarried women, widows, and nuns was considered "robbery" and severely punished.

Defendants could appeal to the emperor (by petitioning an official called the epi ton deeseon) or higher officials with some sort of judicial authority, like provincial governors, the city prefect of Constantinople, the droungarios tes viglas (commander of the city watch, who assumed judicial functions), or judges of the velum (curtain), a college of professional judges who met at the Hippodrome.