Tough question. There is no "right" answer, unfortunately. Oliver Taplin provides a really great and accessible discussion of the topic in chapter 1 of Literature in the Greek World (Oxford 2000). That whole first chapter (at least) is readable on google books. I feel like no one is going to be able to say what needs to be said better than him, in such a small amount of space!

In general, though: Iliad was just one of several epic subjects floating around in the period ca.1000-700 BCE. The Odyssey is another obvious topic, but more generally the nostoi tales ('homecomings') were popular, not only that of Odysseus but also the ill-fated homecoming of Agamemnon, the travels of Menalaus, of Neoptolemos, et cetera. There were also origin stories, like a poem devoted to the birth of Aphrodite. The Iliad and Odyssey in particular seem to represent the masterwork of the "dark age" poet's craft. The stories were modular, right down to the level of individual lines or word groups or individual words. The two surviving epics represent the assemblage of those modular units into larger, cohesive wholes. It is assumed that one person, a true genius, was responsible for this great effort, and that s/he produced work(s) that set them apart from their fellow poets. But that is an assumption. The Iliad and Odyssey were clearly written down at some point, because by the Hellenistic period we hear about written copies of the epics in different places with only minor differences. We also have fragments of the epics on papyri that show minor differences. The question is when then were written down, and under what circumstances.

I think most people these days agree that the epics we have now, with some minor exceptions, are what the poems were like by about 700-650 BCE. That is -- that they were composed (or written or conceptualized or assembled or whatever) by that date, and were not changed significantly thereafter. The two poems have a great deal of internal cohesion, and also some cohesion between them. In that way they seem to be the product of someone(?) with a master plan and an authoritative voice over the whole. Taplin will go into more detail on that and say it much better than I can. Personally I think that by the mid-6th century at the latest, the Iliad and Odyssey had been canonized for the most part and their recital had ceased to be a creative act. By this point, also, value judgments on some kind of basis had already marginalized the other epic archetypes. Some might have existed in written form, but I admit I don't know the specifics on that.

We should also consider the ancient theories of various political figures acting as the "first editors" of Homer's poems who collected all of the written fragments in circulation at the time and commissioned one "official" version for future publications. Plutarch mentions Lycurgus (Life of Lycurgus 4.4), but the most prevalent theory seems to be that Peisistratos was the one who commissioned and circulated the versions of Homer we know today. Cicero mentions this in de Oratore 3.137. Whether or not this was actually the case, and why or how they would have accomplished this, is still being argued by Classicists today. An important aspect of the ancient tradition, however, is that early on the epics of Homer did not exist as a unified whole, but rather as portions or episodes scattered throughout various regions. If you're interested in more details, T. Boyd has written a concise article about the topic, but you need access to JSTOR. I'm sure there is more accessible info, just search for "Peisistratean Recension."