Finding self-reflection in a work of literature has an amateurish reputation and it is, admittedly, very slippery. Nevertheless I can see no way around it: to me, the Iliad and especially the Odyssey are among the most reflexive works of art that I know of.
This is not about the naive question a reader may ask an author: "is that you, in the book"? It is not that simple. But consider an aoidos like Homer: by virtue of his using direct speech in his poetry, in performance he becomes every being, man, woman or god, that he sings about. After all, he is speaking with their voice at that moment. And Homer shows himself very aware of that. Of course this also is the case for every rhapsode performing his works later. But for the composer himself it must have been something new and fascinating, judging from the way he plays with it all the time.
So here the self-reflection starts. For a humorous indication of this, I refer to the Proteus-story (Od 4.384-424) where he uses the (probably old) motif of a shapeshifter daimōn to express the above, in a description of himself, his daily life and the purpose of the Odyssey. For the Iliad, I would like to point out two "complexes" of self-reference: the "exile" and the "Apollo" complex. By complex here I mean a set of connected ideas about a certain topic that pop up everywhere, under many different names. In the Odyssey, we have the "exile returning home", "daughter" and "looking back on the Iliad" complexes.
All I am going to bring forward here is of course unprovable and dependent on a number of assumptions about the poems that certainly not everybody will share. So instead of listing all the individual references of the complexes I prefer to construct a hypothetical background story and leave it to the reader to find the likenesses (or otherwise) as they go through the poems. But first some further explanation is necessary:
If Apollo is the god of poetry and the Iliad is a poem, it follows that there must an intimate relation between the two. It does not mean that Homer sees himself as the god of poetry, but Il 1.8-9 means that the god of poetry brought Achilles and Agamemnon together to quarrel, i.e. it is in the poem that they are brought together.
When the poet speaks as poet, sings rather, it is Apollo who speaks through him.
Apollo is of course the god of poetry, prophecy, healing and archery. All these things are linked together by the poet in a completely consistent way. That the god has these functions, means that his poem also must have these functions. All of them are developed, seriously or humorously, in the Iliad and to a much lesser degree in the Odyssey.
For instance, archery. A lyre is much like a bow: it has to be strung, brought up to the right tension and when plucked it gives off a clear note (ref. the bow-stringing scene on Ithaca). Also, it shoots arrows. This is because a well chosen word is like a feathered arrow: it flies in a straight line into the listeners heart where it may cause, among others, a "pleasurable pain". Hence the ubiquitous phrase "winged words" (feathered rather). And even "wingless words" if they fail to reach their target.
Archers shoot from a distance, unlike spear fighters. This gives them a faint odor of cowardice. This distance also links with an "exile" complex which permeates both poems. An exiled poet also "shoots from a distance" because the true target of his words, his family and the people of his former home town, must always be far away. He must always be "far from the ships" (Il 1.35, 48). The cowardice component is also important because, among a warrior people, not being there to fight with them will always be called that.
The poet sings his songs in select company, the landowner/warrior elite, during a feast-meal (dais) around the wine-filled mixing bowl. Many warriors are killed in his poem, metaphorically he is killing his public during a feast. Reflections of this start in Iliad 1.5 where the killed warriors become "a prey for the dogs and a feast for the birds" and they end in Od 21 and 22 where he goes into full Achilles mode for the last time and kills off his complete audience on a feast day for Apollo.