It's really very uncertain. A couple of Austrian scholars, Georg Danek and Stefan Hagel, have put a considerable amount of effort into trying to deduce some points about the musical side of Homeric performance, and you can listen to the results here -- though I fear Danek's talents lie more in scholarship than in his singing voice (no offence intended, Professor Danek).

Personally, I find it far too speculative to be anything more than a "here's one possibility..." kind of way. The very biggest weakness lies in the impossibility of recovering stylistic devices such as ornamentation. This is not a minor matter. Get a musician friend to take a modern popular song and perform it removing absolutely all ornaments, syncopation, and other devices that wouldn't appear in a Grade 3 piano reduction, then compare it to the original, and you'll see what I mean. A slightly more concrete example: try listening to the vocals in this relatively unornamented performance of a song, then this much more sophisticated (but still fairly low-key) performance. Maybe that'll clarify what I'm saying.

Anyway, the upshot is that there's a good chance that an ancient Greek kitharist performing Homer would have sounded, in terms of the vocals, more like this than like Danek's performance. It's not pleasant to modern first-world ears, but it is at least a real oral epic tradition (recorded in 1930s Bosnia-Herzegovina). Listen for a while to get a feel for the kind of ornamentation he puts in. The language is different from Homer, the musical instrument is different, the rhythms are different, but I think it does give a good feel for the kind of alienness that should realistically be imagined.

The aspects of performance that we can recover are mostly non-musical. We know the rhythm, at least: it was a march rhythm, dominated by a dum-diddy-dum-diddy riff; that much is built into the poetry itself. About the social context of performance, we know nothing until the late 6th century BCE, when competitions in performing Homer were established in Athens: on those occasions, rhapsodes ("song-stitchers") would pick up where the previous one left off. Prior to that, we simply don't know on what occasions Homeric epic would have been performed; from the Athenian competitions, and from one of the Homeric Hymn to Apollo, we can infer that a public performance at a religious festival was one possibility. Other than that we're reduced to speculation. I can list some speculations off if you really want, but I'm reluctant to do so because it might make them look stronger than they are.