I’m not a teacher or professional scholar, but in my amateur and amateurish view, a good place to start would be Homer, Poet of the Iliad by Mark Edwards.
https://www.amazon.com/Homer-Iliad-Mark-W-Edwards/dp/0801840163?keywords=edwards+homer&qid=1540660740&sr=8-1&ref=sr_1_1
Book VI would be a good book to read in its entirety as an entry into the Iliad.
An up-to-date commentary on Book VI, with a useful introduction that summarizes the current state of opinion on the composition of the Iliad (among other things), is in the Cambridge Greek and Latin Classics (Green and Yellow) series by Graziosi and Haubold:
https://www.amazon.com/Homer-Iliad-Cambridge-Greek-Classics/dp/0521878845?keywords=Iliad+Book+VI&qid=1540661527&sr=8-1&ref=sr_1_1
Questions about the origins and composition of the Iliad were placed in an entirely new light, beginning in the 1920s and 1930s, by Milman Parry’s studies of the compositional techniques of Yugoslav oral bards still active in that era, and his demonstration that the Homeric poems show evidence of similar compositional techniques. The question of whether the Iliad was composed in writing or orally had existed since Wolf at the beginning of the 19th century and had been raised even before him, but Parry’s work brought the issue into focus with specificity as to the actual techniques of composition.
This is the revolution I mentioned, which makes the work of 19th century analysts such as Leaf if not wholly obsolete, at least requiring substantial revision and reevaluation. The issue of whether the Iliad was originally written or composed orally or dictated by one or more aoidoi hasn’t gone away (or maybe it’s fair to say it’s been made nearly irrelevant), but Parry’s work completely changed the terms of the discussion by showing exactly and in highly technical detail how a long poem in a difficult meter could be composed orally during actual performance – in real time – using a traditional artificial language built up over time out of metrical fragments that fill specific metrical slots.
You can read about Parry’s work (unfortunately he died, possibly by suicide, before completing a synthesis) in Albert Lord’s Singer of Tales, apparently about to be republished – as long as you’re careful not to take this as the last word, which hasn’t been, and never will be, uttered. There is all too much oral theory nonsense afloat, but at the same time few if any scholars reject it today, and it has had nothing short of a revolutionary impact on Homeric studies.
https://www.amazon.com/Singer-Tales-Third-Hellenic-Studies/dp/0674975731?keywords=lord+singer+of+tales&qid=1540661991&sr=8-1-fkmrnull&ref=sr_1_fkmrnull_1
I should mention that other contributions to an understanding of the composition of the Iliad were also made around the same time as Parry’s work: for example, the recognition that the language of the Homeric poems is an artificial language, not a spoken dialect, and the demonstration of how the poet used “typical scenes” to fill out the narrative.
You can also read West’s book, which Paul cited, reviving some of the 19th century analyst ideas in a new framework that is compatible with oral theory to a large extent. However, I don’t think many Homeric specialists have wholeheartedly embraced his views, which are certainly ingenious and plausible, but in my view go too far beyond what little evidence there is. I think there are other possible, and maybe more satisfying, explanations for the phenomena on which his ideas are based.
Graziosi/Haubold, in my view, treat the issues of the composition of the Iliad with an appropriate level of undecided caution, which I think is where we have to leave the issue unless additional evidence can be found, and that’s highly unlikely. By the way, I see that Barbara Graziosi is coming out with a book on Homer in the Oxford Very Short Introduction series early next year, and I for one will be eager to read it.
One thing that has to be kept in mind is that we don’t really know how the original audiences (or readership?) of the Iliad would have reacted to the poem and its various parts. For example, most everyone today (except scholars who are intrigued by its problems) finds the Catalogue of Ships tedious, but it appears that catalogue poetry was quite popular in archaic Greece. Many aspects of the Iliad that seem puzzling to us may well have played to the Iliad’s original audiences, which we know next to nothing about (along with the circumstances of performance, though I think we can be reasonably certain that in general the Iliad, whenever it may have been reduced to writing, did not reach audiences in written form until the 5th century or even later). Without having reviewed the 19th century literature on the Iliad and its composition in depth, I think, based on what I’ve read about it, that much of it was based on expectations that were derived, consciously or not, from engaging with 18th and 19th century fiction.
Also, would the many small inconsistencies that analyst critics have seized upon have troubled a composer (or a group) working in an oral tradition or audiences enjoying the poem, in whole or most likely in parts, in an oral performance?
Perhaps mwh could suggest some additional reading materials.