Fate is a word whose etymological meaning is "that which has been spoken." For the ancient Greeks, a person's fate was something decreed by three goddesses known as the Fates (Moirai in Greek).
Zeus, as king of the gods, was responsible for making sure what the Fates decreed came to pass. Indeed, even Zeus himself is affected by fate even though he is the king of the gods. This is evidenced by what happens to his son Sarpedon in Iliad 16. When Sarpedon is about to be killed by Patroclus in Iliad 16, Zeus considers rescuing him from the battle until Hera reminds him that he should not "save a mortal from the pains of death, one long since doomed by fate."
Of course in the Iliad, the person whose fate we hear the most about is Achilles. In Book 1, his mother Thetis reminds Achilles of his fate:
if only it were your fate to stay by the ships, free of pain and sorrow; but you, more wretched than other men, must meet an early death (A.S. Kline's translation)
Achilles, however, does not remain in his tent by the Greek ships. The death of his best friend Patroclus in Iliad 16 causes him tremendous pain and sorrow. To avenge his friend's death, Achilles kills Hector, who, as he dies in Iliad 22, also reminds Achilles of his fate:
But think, lest the gods, remembering me, turn their wrath on you, that day by the Scaean Gate when, brave as you are, Paris kills you, with Apollo’s help. (A.S. Kline's translation)
Unfortunately for modern audiences, we do not get to see how fate plays out in Homer's Iliad, because this epic ends with the funeral of Hector. The death of Achilles would be told in another poem that was part of the so-called epic cycle.