The word "climax" is defined by the Oxford English Dictionary as:
The culmination, peak, or apex of something; the most important or exciting part of a film, contest, etc., usually happening near the end.
In literature, the climax is the point at which the narrative tension reaches its highest pitch and can no longer be sustained. The tension is often released by some kind of crisis, defined as "a vitally important or decisive stage in the progress of anything; a turning-point," which forces the action to a resolution.
The Iliad begins with "the rage of Peleus' son Achilles," and climaxes when that rage results in the death of Hector, prince of Troy.
The poem starts at the end of a very long war, in which neither side has made any significant progress for some time. All the participants are tired and disaffected. A minor disagreement between Achilles and Agamemnon turns into a major falling-out with huge ramifications, as Achilles, foremost among warriors, withdraws from the battlefield. The Achaeans lose their best man and their morale is badly damaged.
Meanwhile the Trojans, led by Hector, take notice of Achilles' absence. They have been under siege for a decade, and unable to throw off the Achaeans largely due to Achilles' prowess in battle. With Achilles no longer in play, Hector sees an advantage for the Trojans and presses it for all it's worth. The war, which has been at a stalemate, comes roaring back to life, and the Achaeans are beaten back from the walls of Troy as far as their own ships. When the Trojans set the Achaean ships on fire, trapping the soldiers on the shoreline, Achilles' dearest friend, Patroclus, joins the battle, dressed in Achilles' armor in an attempt to rally the troops and fend off the Trojan advance. Hector kills Patroclus in combat.
Achilles is devastated. His anger towards Agamemnon forgotten, he reenters the battlefield and cuts a bloody swathe across it, furious for revenge. The Achaeans take courage from Achilles' presence and throw the Trojans back, killing hundreds. The Trojan army retreats behind the city walls—all except Hector, who stands before Troy's gates and waits for Achilles to come for him. His father, King Priam, begs him to come within the walls, saying Achilles has already killed so many of Hector's brothers; he, Priam, cannot stand to lose another son. Hector refuses to yield, however, and holds his ground until Achilles is almost upon him, determined to kill Achilles or be killed by him rather than retreat:
...Better by far for meto stand up to Achilles, kill him, come home aliveor die at his hands in glory out before the walls....So [Hector] wavered,waiting there but Achilles was closing on him nowlike the god of war...
But as Achilles draws closer, Hector's nerve fails, and he takes off running:
Hector looked up, saw him, started to tremble,nerve gone, he could hold his ground no longer, he left the gates behind...
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and away he fled in fear andAchilles went for him, fast, sure of his speedas the wild mountain hawk ...breakneck on in furywith Hector fleeing along the walls of Troy,fast as his legs would go.
... they raced, one escaping, one in pursuitand the one who fled was great but the one pursuinggreater, even greater-their pace mounting in speed since both men strove, not for a sacrificial beastor oxhide trophy, prizes runners fight for, no,they raced for the life of Hector breaker of horses.
All the action of the war these past ten years is now honed to the knifepoint of these two men, Achilles and Hector, confronting each other at last. They are the proxies of their respective armies, Achaean and Trojan, and the outcome of their confrontation will determine the final course of the entire war. These men are individually more important than all the other characters in the poem; the gods' interference in the war is centred largely around the two of them. This climactic scene of Achilles running towards Hector before the city gates is a depiction of an unstoppable force meeting an (almost) immovable object—the results, more than climactic, are cataclysmic. Achilles catches up to Hector and kills him savagely, lashing Hector's corpse to his chariot and dragging it around the walls of Troy to the horror of the citizens watching from above. The spirit of the Trojan army is destroyed.
The final few books of the poem comprise the denouement, describing how Achilles' rage gives way to grief for Patroclus, which allows him to empathise with the grief King Priam feels for his dead son, Hector. Achilles returns the body of Hector to the Trojans, and the poem ends.
(All quotes are from the Robert Fagles translation of the Iliad.)
In my opinion, the climax of the Iliad comes in Book 22. I think that the climax comes when Achilles kills Hector. The reason that I say this is because this is the apex of the rage of Achilles.
I think that much of the point of this epic is the anger of Achilles. After all, the poem starts by asking the goddess to sing of the wrath of Achilles. We see Achilles' anger cause him to withdraw from the war and sulk in his tent because his pride has been offended. Later, we see his anger cause him to fly into a murderous rage as he takes revenge for the death of Patroclus (which was caused, you can argue) by Achilles' anger.
So the whole epic is about the consequences of Achilles' anger. This anger peaks as Achilles kills Hector and gets his revenge. After that, the rest of the epic is really just about tying up loose ends caused by Achilles' anger.
The action of the Iliad leads to the high point, or climax, of the poem in which Achilles kills Hector. This is such a fraught moment because it is a clash between each side's greatest warriors, and the outcome could turn the tide of a years-long war.
As the poem opens, Achilles is sulking in his tent and refusing to fight because Agamemnon, his commander, has taken his war prize, Briseis, as his own. Achilles's act of withdrawal sets up a chain reaction. Because Achilles is not fighting, the Trojans, helped by the gods, begin to win the war. As a response, Achilles's good friend Patroclus asks if he can wear Achilles armor into battle. This way, it will look like Achilles is back, inspiring the Greeks and intimidating the Trojans. Achilles agrees. During battle, Hector kills Patroclus. This so enrages Achilles, who loved his friend dearly, that he reenters the war. He kills so many Trojans that the river is clogged with their bodies, and the gods have to intervene to slow Achilles down.
Finally, Achilles confronts Hector. Hector knows he cannot win, and he wants to flee, but the gods decide it is Hector's time to die and persuade him to fight. At the end of this climatic scene, Achilles kills Hector, avenging Patroclus. He attaches Hector's corpse to the back of his chariot and races around the walls of Troy, dishonoring the fallen warrior's body. With their most fearsome adversary gone, the tide seems to have effectively turned in the Greeks' favor.