The short answer is that we probably don't know - certainly not in any detail. There's very little substantial information about the buildings on the Palatine after the end of the Severan dynasty (AD 235), only the occasional scrap or passing reference. The archaeology isn't always a great help either. Past excavators, here and elsewhere in the city of Rome, have tended to plunge straight down to the monumental and exciting stuff in the early imperial or republican layers, ignoring or destroying evidence of later periods. Why bother carefully recording sonme grotty 5th or 6th century apartment blocks when you could be hunting for Romulus' hut, right?

So, what do we know? After Constantine, the hill seems to have been occupied by private homes and other buildings. The Regionary Catalogues - lists of buildings and landmarks written sometime in the early 4th century - list 89 larger homes and over 2000 apartments on the hill. Emperors did still occasionally visit in the 4th century, most notably Constantius III in 356, so it would be fair to assume some kind of skeleton staff remained to make sure the palace would be ready if someone showed up. The Gothic kings Odoacer and Theoderic seem to have lived there too, and according to Platner and Ashby some brick stamps of Theoderic have been found, suggesting he engaged in some rebuilding. Several churches were built on and around the hill from the 4th century onwards, including San Sebastiano, which was constructed on the former site of the temple of Elgabal, the Syrian deity brought to the city by the emperor Elagabalus.

For further reading, check out the Platner and Ashby link above, and also Richardson's New Topographical Dictionary of Ancient Rome.