In Ancient Rome, who would act as the police detectives? Who would try to figure out who’s guilty for murders so they could have a trial? What methods for investigation would they use?

There have been a few previous answers that might be helpful here:

How easy was it for fugitives to evade capture in the Roman Empire? by u/mpixieg

How were crimes investigated in Roman times? by u/AwesomeDog59

And a couple by me:

Prior to DNA evidence, finger prints, etc. how did they solve murders and actually know if they convicted the right person?

I'm living in Ancient Rome and I just murdered someone. What chances do I have of getting caught?

In short - no one! If someone got murdered, that was a problem for the murdered person's family. If they could bring the murderer to court, then the government would deal with it. If not, then there was nothing to investigate. In Rome, the praetor was in charge of investigating, if someone was brought to him. But the government did not investigate crimes on its own.