Locusta often comes up in these discussions but I don't think that is a good way of answering. For one, she wasn't a serial killer, she was a poisoner or an assassin. So even if she was responsible for multiple deaths it isn't falling under the standard definition of serial murder. Secondly she is, in a word, a tad overhyped. The only deaths that can be attributed to her are those of Claudius and his son Britannicus. She seems to have had a reputation before but, again, there is no reason to think of her as a serial killer instead of an assassin.

In mythology you can find things like the bandits Theseus killed, most famously Procrustes, who would tie his victims to a bed and cut off any part that did not fit. Of course this is mythology and not history but may reflect a certain angst over sadistic bandits.

Might be stretching the definition of "serial killer" here, since it was more of a conspiracy that killed a large number of people, but I'll repost a comment I made a few years ago about a story from Livy:

Livy relates a story in the year 332 BC about a conspiracy of patrician women poisoning their wealthy husbands by the dozens. The deaths were thought to be the result of some unknown plague, until a slave told the curule aedile, Quintus Fabius Maximus, and the source was discovered.

Source

For a slightly easier read, here's a translation provided by Steven Saylor in his book Roma:

This year gained an evil notoriety, either because of pestilence or human guilt. Since the authorities are not unanimous on the point, I would gladly believe it was disease, not poison, that carried off so many victims. But lest I impugn the credibility of our sources, I shall relate the sordid details just as they've been handed down to us.

The foremost men in the state were being attacked by the same mysterious malady, which in almost every case proved fatal. A maidservant went to the city magistrate, Quintus Fabius Maximus, and promised to reveal the cause of these suspicious deaths, provided the state would guarantee her safety. Fabius went at once to the consuls, who referred the matter to the senate, which authorized a promise of protection and immunity.

The maidservant then accused certain women of concocting poisons. If officers would follow her at once, she said, they could catch the poison makers in the act. The officers followed the informant and did indeed find the accused compounding poisonous substances, along with batches of poisons that were already made up.

The evidence was seized and brought into the Forum. Twenty high-born matrons, at whose houses poisons were discovered, were brought before the magistrates. Two of the women, Cornelia and Sergia, both from ancient patrician families, contended that the concoctions were medicinal preparations. Accused of lying, the maidservant suggested that the women should drink some of the medicine themselves, if they wished to prove it was harmless.

The court was cleared of spectators. The accused women consulted among themselves. All consented to drink the potions, whereupon they all died.

Their attendants were arrested at once, and informed against a large number of matrons. Eventually, 170 women were found guilty. Up to that time there had never been a public investigation of poisoning in Rome. The whole incident was regarded as an evil portent, and the women were thought to have acted out of madness rather than deliberate wickedness.