This depends on your status. If you're part of the elite you almost certainly have heard Homer being recited and know parts by heart. I suspect that even if you were not part of the elite you'd still know the basic story - Homer's account of the Trojan war wasn't an isolated thing and that backdrop was used by playwrights and featured in many myths.

Originally the poems were recited by poets who'd alter the story to suit themselves and only when it was written down (probably during the archaic period) would it start to have some fixity. I believe the version we have now comes from copies made at the Library of Alexandria. So there may have been many other versions of Homer floating around out there. You'd probably have been exposed to at least one of those versions.

It seems the ancient Greeks had a pretty good ear at remembering things - we see in the plays of Aristophanes that he's constantly parodying other tragedies and dropping quotes from other plays. That would only make sense if his audience (the broad spectrum of Athenians) would get those references. So it stands to reason that yes you would be familiar with his works.