The introduction by Michael Grant to the Penguin edition of Cicero - Murder Trials (Penguin, 1990) gives a concise description of how a murder trial went on during the late Republic (so not quite Pax Romana, depending on the definition, but still ancient Rome). Cicero among other things, also functioned as an court advocate, mainly for the defence, and some of his extant writings are his court speeches in some famous murder trials. These are some of the main sources for court procedure at that time.

From the time of Sulla, court judges to the 7 or more standing courts in Rome at the time, was drawn up from the around 450 qualifed senators in the Roman Senate (of 600). "75 [judges] in each of the more important courts and 51 in the rest". The judges were selected through "a complicated process of rejection and counter-rejection in which both the prosecution and defence took part; though, even so, corruption was by no means eliminated" (p. 14-17):

Rome possessed nothing resembling a public prosecutor or attorney general; and it was not the duty of any state official to initiate prosecutions. Every citizen was at liberty to make a charge against another, and then the court's chairman, if he thought there was a prima facie argument - or was advised in this sense by a committee of his judges - could put the case down for trial [...] It was the responsibility of the two parties concerned to produce the evidence, though the accuser might have the assistance of state officials in preparing it. On the other hand the accused was given impressive opportunities for his defence. He might, on occasion, have as many as six counsel, and they were normally allotted half as much speaking time again as the prosecution [...] A trial included not only the set speeches of the advocates, but sharp cross-examination as well. The latter have not survived, but their substance is often incorporated, in continous form, in the published (and amended) versions of Cicero's speeches that have come down to us. The judges listened in silence, and were forbidden to speak to each other. Finally, they voted - guilty, or not guilty, or (as in Scotland) not proven. The chairman did not vote, but pronounced judgement and sentence, to which the traditional Roman right of appeal did not apply. The carrying out of the punishment was likewise his responsibility.

As mentioned, the extant speeches by Cicero are only part of the trial proceedings, but I can really recommend them, as they are not only interesting history, but make for some surprisingly suspenseful crime literature as well, and they provide the reader with some insight into the brutal and corrupt side that often was the outcome of Roman patron-client system and the accompanying party politics. Unfortunately we lack the trial verdicts for several of the cases.

Edit: Some typos.