I think that there is a complex view of immortality offered in the Iliad. There are aspects of characterizations where the pursuit of immortality is evident. Achilles recognizes that there is a particular immortal glory in his participation in the war against Troy. He understands that glory in war carries with it an immortality that is worthy of pursuit. At the same time, Hector understands that defending his father's city will achieve immortality as it secures “winning my father great glory.” While Hector is not motivated by immortality, he understands that it is a part of the pursuit in war. As an exercise, war is a vehicle in which immortality can be achieved. It is in these realms where the pursuit of immortality is evident.
Yet, what makes the Iliad so profound is that it questions such pursuits of immortality. If war is the ultimate testing ground for immortality, the brutality shown causes one to question such a condition. Homer constructs a realm in which human mortality is seen as intrinsic to being in the world. Any pursuit of immortality is one that has to be seen as impossible for humans to achieve for it is the realm of the divine:
Thereafter beginning from the left he poured drinks for the other
gods, dipping up from the mixing bowl the sweet nectar.
But among the blessed immortals uncontrollable laughter
went up as they saw Hephaistos bustling about the palace.
Thus thereafter the whole day long until the sun went under
they feasted, nor was anyone's hunger denied a fair portion,
nor denied the beautifully wrought lyre in the hands of Apollo,
nor the antiphonal sweet sound of the Muses singing.
The divine are those who enjoy immortality. It is reserved for the Greek Gods and Goddesses. Human beings who wish to achieve immortality are establishing a foundation for futility and hurt, something that Agamemnon relays to Nestor:
Aged sir, if only, as the spirit is in your bosom,
so might your knees be also and the strength stay steady within you; but age weakens you which comes to all; if only some other
of the fighters had your age and you were one of the young men!
"Age weakens" all human beings. The pursuit of immortality is understood within this context. The inevitability of age and the ravages of time can never be evaded. Warriors on the battlefield who strive for immortality find it undercut by such a reality. This becomes one of the most important elements regarding the pursuit of immortality that Homer offers.