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".