The Iliad starts at kind of an odd spot. It starts by describing the argument between Agamemnon and Achilles (Akhilleus in my translation) over the women they have acquired from raids. It mentions off-hand that they have been there for nine years. Why is nothing about those nine years described at the beginning to introduce the story?
Whether the Iliad starts in the middle depends on what the subject of the poem is. If the subject is the whole Trojan War, then certainly the poem starts in the middle, and finishes well before the end. But is that really the subject? The opening line of the poem is:
μῆνιν ἄειδε θεὰ Πηληϊάδεω Ἀχιλῆος
That is, “sing, goddess, the rage of Peleus’ son Achilles”. If the subject of the poem is the rage of Achilles (and μῆνιν, rage, is the very first word) then the Iliad begins at the beginning, with the cause of Achilles’ rage (the quarrel with Agamemnon over the captive Trojan women Chryseis and Briseis), continues with its consequences (Achilles sulks; the battle goes poorly for the Achaeans; Patroclus impersonates Achilles and is killed by Hector), and ends with its resolution (Agamemnon returns Briseis to Achilles; Achilles kills Hector; Priam begs the return of his son’s body).
There was a whole cycle of poems detailing the Trojan war and its aftermath. See Wikipedia. Of these, only the two attributed to Homer have survived intact.
So you could think of it as starting in the middle, or you can think of it as being one installment in a long series, most installments of which have now been lost.
Much of the audience would have known the outlines of the whole story already, so there was no need to start it with a summary of "what has gone before".
The existing answers ignore the most important comment from Antiquity on this matter, namely chapter XXIII from Aristotle's Poetics (in Ingram Bywater's 1898 translation), quoted below, or in S. H. Butcher's 1922 translation). Although Poetics (or at least what has survived of it) deals primarily with drama, especially tragedy, it also contains comments on epic poetry.
Aristotle writes (emphasis mine):
As for the poetry which merely narrates, or imitates by means of versified language (without action), it is evident that it has several points in common with Tragedy.
The construction of its stories should clearly be like that in a drama; they should be based on a single action, one that is a complete whole in itself, with a beginning, middle, and end, so as to enable the work to produce its own proper pleasure with all the organic unity of a living creature. Nor should one suppose that there is anything like them in our usual histories. A history has to deal not with one action, but with one period and all that happened in that to one or more persons, however disconnected the several events may have been. (…)
If Homer's Iliad had recounted the entire story of what is now known as the Epic Cycle, it would have been more like history than epic poetry. Aristotle's view on this is different than that of modern readers who are used to reading epic fantasies that span multiple volumes (Lord of the Rings, A Song of Ice and Fire). We also shouldn't forget that the Iliad has its roots in oral literature.
Herein, then, to repeat what we have said before, we have a further proof of Homer's marvellous superiority to the rest. He did not attempt to deal even with the Trojan war in its entirety, though it was a whole with a definite beginning and end—through a feeling apparently that it was too long a story to be taken in in one view, or if not that, too complicated from the variety of incident in it. As it is, he has singled out one section of the whole; many of the other incidents, however, he brings in as episodes, using the Catalogue of the Ships, for instance, and other episodes to relieve the uniformity of his narrative.
Aristotle appears to approve of Homer's approach to the material around the Trojan War. (Aristotle also praises Homer for other aspects of his work.)
In summary, the Iliad starts "in the middle" if you take the point of view that the entire story of the Trojan War should be told, possibly from Helen's abduction by Paris (if it was actually an abduction) to the Fall of Troy. But Aristotle considered it as a self-contained story.