To answer this I should start with a quote:

“During the Roman republic murder was not a crime…republican Romans had neither the capacity nor the inclination to make the essentially private act of malicious and intentional homicide an offense actionable by the government…The Romans not only had no legislation prohibiting murder, they had no word for murder.”

(Judy E. Gaughan, Murder Was Not a Crime: Homicide and Power in the Roman Republic (University of Texas Press, 2010), pg. 1-3)

Sounds like you could just go around murdering people whenever you wanted! But that was obviously not the case. The Romans did think that murder was a crime, but they didn’t have a police force, and the state was not a legal person like it is today. The government couldn’t act on its own accord to prosecute anyone for a crime committed by one private citizen against another private citizen.

If you murdered someone, your chances of being caught depended entirely on another citizen accusing you. The murdered person’s family would be responsible for bringing an “action” to the court. The person in charge of accepting or rejecting actions in ancient Rome was the praetor, who would then appoint a judge to hear the case. Then it was up the dead person’s relatives to convince the judge that you were the murderer - they or someone else saw you do it, you had the motive/opportunity, you had a weapon, bloody hands, etc. If the dead person didn’t have any family, or anyone else to vouch for them in court, then your chances of getting caught were zero.

Essentially “The system as a whole seems to have been designed to work best as a state-sponsored form of arbitration” (Andrew M. Riggsby, Roman Law and the Legal World of the Romans, Cambridge University Press, 2010, p. 116). The case could be settled with a fine or some other mutual settlement, or for a particularly bad case maybe you could be exiled or forced to work in the mines. The Lex Cornelia, for example (issued by Sulla in 81 BC) said that anyone convicted of murder should be exiled. But there was no state prison, and the state typically couldn’t execute you for a crime against some other citizen.

The exception to this was if you killed your own relative, your father especially, but any other family member too. Under the Lex Pompeia de parricida (issued by Pompey sometime in the 1st century BC), you would be “sewed up in a sack and thrown into the sea” (maybe with a monkey tied in the sack with you...just to make things more interesting). Later in the imperial era you could also be burned to death. But still, someone else would have to bring you before the court first.

It didn’t work the other way around though, because as paterfamilias, a father always had the right to kill any family member for whatever reason. There were also some other occasions when killing someone was perfectly legal. The earliest laws in the Twelve Tables (about 450 BC) don’t say anything about murder as a crime, but you were allowed to kill a thief breaking into your house at night (Table VIII). According to Augustus’ Lex Julia, a father could legally kill a man committing “adultery” with his daughter. The husband could also kill the adulterer, but not his wife, although if he did kill his adulterous wife he probably wouldn’t be punished.

Unlawful killing is mentioned in the laws most often in the context of slaves killing or being killed. If you killed someone’s slave, the slave’s owner couldn’t bring an action against you for killing the slave (because the slave was not a legal person), but it was considered damage to property, so you would be liable for reimbursing the cost of the slave/the slave’s labour. This was part of the Lex Aquilia, dated to the 3rd century BC.

Your chances of being caught if you weren’t a citizen were much higher, especially if you were a slave, and especially if you killed your master. That was an unthinkably bad crime for the Romans. Unlike a citizen, you would definitely be caught, tortured, and killed. (Torturing slaves was a really important part of the process for some reason.) I'm not sure what happened if you were a freedman, but freedman could be re-enslaved if they committed other crimes.

So in brief: the state arresting you and punishing you is a very modern concept. If you murdered someone, it was up to that person’s relatives to bring you to court; if they didn't, you could probably get away with it.