In general, stories of demonic possession seem to come from the Judeo-Christian tradition, at least in the religions of the Mediterranean, and so don't really feature before the rise of Christianity. Hauntings are a different thing, and I'll come to those in a bit. The first thing to point out however is the source of these occurrences. Unlike Christianity, Graeco-Roman paganism didn't really have the concept of an evil being such as Satan, with demons and other beasties that did his bidding in the world. Sure there were spirits and supernatural creatures that humans should be scared of, who could do very nasty things if you got on their bad side, but they weren't essentially 'evil' in the way that the devil of Christianity is. The main type of supernatural force involved in haunting or attacking people in Graeco-Roman paganism is the spirits of the dead.

From very early on, the spirits of those who died violently or before their time were thought to hang around, unable to properly rest until they had been given a proper burial. We have evidence of this belief in Homer (7th century BC, maybe earlier), as the ghost of Patroclus appears to Achilles in the Iliad (23: 62-76):

Sleep poured around him in sweetness, seized Achilles and released the concerns of his heart. For he had worn out his glorious limbs by chasing Hector to windy Ilium. The ghost of unfortunate Patroclus came upon him, resembling the living man in all things, in stature, in his beautiful eyes, and in his voice, and it was dressed in clothes resembling his. The soul stood over his head and spoke to him: “You sleep, but you have forgotten me, Achilles! You cared for me when I was alive, but not now I’m dead. Bury me as quickly as possible, so that I may pass through the gates of Hades. The souls ward me back a distance, the ghosts of the dead, and they do not yet allow me to mingle with them beyond the river, but in this condition I wander about the broad-gated house of Hades. Give me your hand, I beg you through my tears. For I shall not come again from Hades, when you have given me the fire that is my due.”

Even the spirits of those who had been properly buried could come out of their tombs if they weren't properly looked after, as described by Ovid (Fasti):

But once, waging a long war with fierce weapons, They neglected the Parentalia, Festival of the Dead. It did not go unpunished: they say from that ominous day Rome grew hot from funeral fires near the City. I scarcely believe it, but they say that ancestral spirits Came moaning from their tombs in the still of night, And misshapen spirits, a bodiless throng, howled Through the City streets, and through the broad fields. Afterwards neglected honour was paid to the tombs, And there was an end to the portents, and the funerals.

This story comes when he's talking about the Parentalia and Feralia festivals (celebrated in late-February), at which Roman families would go to the graves of their dead relatives and offer food and wine to keep them happy. It's clear from Ovid's warning what could possibly happen if they were neglected! There's another festival that's relevant for my answer to your question: the Lemuria (mid-May). Again, the description of the rituals comes from Ovid's Fasti:

When midnight has come and lends silence to sleep, the worshipper who bears the olden rite in mind and fears the gods arises; no knots constrict his feet; and he makes a sign with his thumb in the middle of his closed fingers, lest in his silence an unsubstantial shade should meet him. And after washing his hands clean in spring water, he turns, and first he receives black beans and throws them away with face averted; but while he throws them, he says: ‘These I cast; with these beans I redeem me and mine.’ This he says nine times, without looking back: again he touches water and clashes Temesan bronze, and asks the shade to go out of his house. When he has said nine times, ‘Ghosts of my fathers, go forth!’ he looks back, and thinks that he has duly performed the sacred rites.

The ritual here is aimed at protecting the house from any ghosts that might be lurking there, perhaps dead family members, but certainly not just them exclusively. This was also not the only method for protecting the house from haunting or attack by aggressive spirits. Apotropaic wind chimes known as Tintinnabula may have been hung around the house, and some homes in Pompeii had paintings of Phallic figures like Priapus near doorways.

The restless spirits of the dead could be manipulated by witches and other magical practitioners, usually for divination but sometimes to attack other living people. My post that /u/hillsonghoods linked has a story from Apuleius' Metamorphoses, in which an old witch raises a ghost to attack a miller on the orders of the miller's wife. This doesn't result in a possession or a haunting, but the death of the miller, who is found hanging from a noose after his slaves break a door down. Interestingly, his ghost then goes off to his daughter's house and tells her that about her step-mother's involvement in his murder.

So, to sum up. The ancient Greeks and Romans didn't really have stories of demonic possession, but they did have hauntings and ghost stories. The spirits of the dead in particular could rise from their graves and attack the living, and so various rituals and protective objects were put in place.

Further reading:

Graf, F. (1997). Magic in the Ancient World (F. Philip, Trans.). Cambridge (Mass.) and London: Harvard University Press.

V. Hope. 2007. Death in ancient Rome a source book

Johnston, S. (1999). Restless Dead: Encounters between Living and Dead in Ancient Greece. Berkeley: University of California Press.

Luck, G. (2006).Arcana Mundi: Magic and the Occult in the Greek and Roman Worlds. Baltimore: John Hopkins University Press.

Ogden, D. (2001). Greek and Roman necromancy. Princeton ; London: Princeton University Press.

Ogden, D. (2002). Magic, witchcraft, and ghosts in the Greek and Roman worlds: A sourcebook. Oxford: Oxford University Press.

There's definitely more to be said, especially because this is slightly orthogonal to your question, but an excellent comment from a couple of months ago from /u/bigfridge224 about the way that Romans perceived witches, which might shed some light on pre-Christian Roman attitudes to dark magic - hope that helps!