The genres of the Iliad are: Epic Poetry, Tragedy and War Drama
This ain't just any old epic: it's the epic that made epics epic.
The Homeric poems (the Iliad and the Odyssey) are epic, because our concept of epic comes from Homeric poems. If that sounds too circular, then just bear in mind that the Iliad is an extremely long narrative poem, which deals with the heroic actions of mortals, gods, and demi-gods. For the Ancient Greeks, it was also important that an epic be written in the poetic meter of dactylic hexameter—which the Iliad is.
At the same time, however, the Iliad is also a tragedy, because it focuses on the downfall of a great hero (our boy Achilleus) as a result of his own flawed character. In this case, the problem is a three'fer: his super-excessive anger, pride, and grief.
Because most of the Iliad depicts battles in the Trojan War, it also falls into the category of War Drama. As such, it provides many important insights into the nature of war and its place in human life (and human death—hey-o!).