A lot of the city became abandoned. During the Gothic Wars and the several subsequent sieges of Rome, the city became virtually empty of citizens. The Gothic army sent many of the high ranking Romans to their latifundias outside the city and kept them as hostages of some type. The instability of the city (during the Gothic war alone there were 3 separate sieges, which forced the tiny population to live off of plants and what they could find around the city) caused people to move to more stable areas. Rome had long been simply a spiritual center of the Empire - political power shifted to cities like Mediolanum (Milan) and Ravenna.

The first time he captured Rome, Totila began to destroy parts of the city. It was only when the Roman General Belisarius wrote him an impassioned letter (which we still have), urging him to think about what his reputation would be throughout the ages if he continued on his current course, that he stopped.

By the middle ages, people that remained in the city were clustered in tiny pockets of the city. Cows were grazing in the Coliseum and the ancient monuments and temples were falling into disrepair. So, to answer your question, you probably could just claim a villa as your own. There wouldn't be many people to stop you, but you would have a hard time finding one that was intact enough for your use that wasn't (likely) already occupied.

I've suggested this book to a few people lately concerning what it was like in Rome after the fall, The Gothic War by Jacobsen is a fantastic look at life in Italy immediately after the fall of the western empire all the way up to the end of the Gothic War (554). Also, The Ruin of the Roman Empire by James O'Donnell is a great read, but it covers a longer time period than just the period your question concerns. Both of these are excellent sources and are where a lot of this information comes from.

Edit: Spelling. And, I thought I'd also add that a lot of structures in Rome were used for building material. For example, if you look at the coliseum today, you'll notice holes all around it. These once held bronze clasps which were removed and re-purposed. Stone and marble from temples, basilcas, etc. was used similarly. So you could potentially end up living in a home made from a Roman monument.