So, first I'll define terms. A clear definition (from wiki) of a serial killer is "a person who murders three or more people, usually in service of abnormal psychological gratification, with the murders taking place over more than a month and including a significant break (a "cooling off period") between them". The number of victims apparently can fluctuate in the definition, the basic premise being a person who makes a habit of murdering numerous people over a period of time for some sort of psychological gratification. So that's actually pretty broad.

There is a tradition in Roman literature of female "poisoners." Examples include allegations that Augustus was poisoned by Livia, or Claudius by Agrippina. However, this is probably just slander, informed by a masculine literary tradition particularly hostile to the idea of females exercising political power. However, that doesn't mean this sort of thing never happened. A prominent example of a poisoner who got quite a bit of "airtime" in the ancient sources is Lucusta, a Gallic poison-mixer who was retained for her expertise in murder by the Julio-Claudians (Tacitus, Annals 12.66, 13.15). As she was apparently well known, and a professional killer, I'm not quite sure if you'd call her a serial killer as opposed to a sort of assassin (she presumably acted for money as much as any gratification), but she's worth a mention.

The Greco-Roman literary tradition has a long and proud history of criticising the abuse of power, especially with violence. The Romans believed that the Republic was created due to a rebellion against the kingship due to the rape of an aristocratic woman by the king's son, for instance. The Greeks had a word for it, hybris (hubris). That tyrants killed people for their pleasure was, aside from no doubt being a common occurrence with unpleasant leaders, was also a prominent literary theme for many writers, whether it be Livy or Cicero. Note that all Roman citizens in the Republic had the right of provocatio, appeal to a tribune against the decision of a magistrate, for instance the death penalty, for precisely this reason.

A good example of a powerful man abusing his power and killing people for his enjoyment, which may very well qualify as "serial killing", is Gaius Verres as depicted in Cicero's lashing condemnation of the man in a legal trial (In Verrem). Verres was the Roman propraetorian governor of Sicily, and his conduct there was so infamous that Cicero, always eager for public attention, brought him to trial. Among many serious crimes, Cicero accused Verres of flogging and crucifying a Roman citizen (hence the poor man's cries of civis Romanus sum!), and the random fatal bashing of an elderly Roman man. Verres was also accused of other unjust killings of non-citizens, but it was the murder of Roman citizens that would really have rustled Cicero's audience. His speech was so effective that Verres' lawyer, himself known as a great advocate, gave no reply and instead advised his client to flee Rome. Years later he was killed under the orders of Marc Antony.

If you included the abuse of power to kill people arbitrarily as a form of serial killing, then you could find numerous examples in Greek and Roman literature, though I have already mentioned the best examples that come to mind at the moment. The Greeks were particularly sensitive to hybris so if I were more familiar with my Greek literature I'd no doubt have more examples.

EDIT: I know OP and probably most readers would be asking more for examples like Jack the Ripper or Elizabeth Batharoy. I can't think of anything like the former unfortunately. However, note with Elizabeth for instance the abuse of power inherent in her crimes. This is a theme that is quite similar to what I have been talking about and was prominent in ancient literature. However, regarding a "common" man like The Ripper, the ancient writers I'm familiar with tended to write about politics, power and war and not much else: the actions of a random slasher in downtown Subura could only have appeared in such accounts if it had political ramifications, which of course it would not unless somebody important were targeted. The idea of a commonor killing an aristocrat was of course repellent to the Romans (see the humiliating death of Commodus at the hands of a wrestler), and so if such a thing occurred I would expect it to have been recorded. People who killed for fun in a dangerous place like urban Rome no doubt existed, but due to the focus of the writers on power, politics and war such events, even if they were the talk of the city among the poor, they would likely not ping the radar of the elite who wrote and consumed the literature we have today.

*"[but] I am a Roman citizen!" A Roman citizen was not liable to the penalty of crucifixion unless he was a deserting soldier or some other brand of military traitor, and even so he had to be ritually stripped of his citizenship first (mostly by beating...).

Postscriptum: That story of Crown Prince Sado is a cool one, and there is apparently a bit of debate about the story's authenticity, just as there is with the criticism of Verres by Cicero. But in my view, with these sorts of detailed contemporary narratives, where there's smoke there's fire, so we can enjoy the story without worrying about being completely misled.

Thanks for the input. I find it truly fascinating that there seems to be no records of serial killers in the ancient world.

I think that you hit the nail on the head. I think life back then was so full of everyday horrors that a serial killer would go unnoticed, or its just not a big deal. What stirred the design to ask my question was when I was watching "From Hell" and I was thinking about the hypothesis that Jack the Ripper and H H Holmes were the same person. I thinking is really wild that those two are considered, from my knowledge, the first serial killers in recorded history. People with those types of disorders must have existed in the ancient world.