What comparison can you make between Macbeth's and Achilles's death?

Achilles’s death is not actually part of Homer’s Iliad, but the commonly accepted version of the story is that Paris shot an arrow into Achilles’s heel, the undefeated warrior’s only physical weakness. However, Achilles’s death is mentioned in Book 22 of the Iliad—right as Achilles has mortally wounded Troy’s fiercest fighter and Paris’s brother, Hector.
Hector asks that his body be delivered back to Troy so that he might have a proper burial. When Achilles refuses Hector’s dying request, Hector replies:

I know you truly now, and see your fate, nor was it mine to sway you. The heart in your breast is iron indeed. 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 (Book 22, A.S. Kline translation 2009).

Hector essentially predicts Achilles’s fate, but in order to compare Achilles’s death with Macbeth’s, you have to understand the reason behind Achilles’s demise.
It could be argued that Achilles’s desire for revenge on Hector for killing his friend Patroclus caused him to violate sacred rules about the proper treatment of an enemy’s corpse. In fact, Achilles even says he wishes he were angry enough to cannibalize Hector’s body—which Achilles says to Hector as he lies on the ground dying. This shows that Achilles was in the wrong for what he did to Hector’s corpse.
In addition, Achilles’s hubris causes him to ignore Hector’s warning. His prowess as a warrior coupled with his lust for revenge blind him to the possibility that he will die at Troy.
As for Macbeth, he possesses a similar hubris. His thirst for power, and willingness to do whatever it takes to keep it, causes him to overlook the witches’ prophecies in act 4, scene 1.
The first spirit that appears to Macbeth from the witches’ cauldron warns him to fear Macduff. However, the second spirit tells him not to fear “for none of woman born / Shall harm Macbeth” (4.1.83-84). The third spirit says that no man can hurt Macbeth “until Great Birnam Wood to high Dunsinane Hill / Shall come against him” (4.1.96-98).
Macbeth immediately deems the first prophecy irrelevant because he thinks the second and third are impossible. Therefore, he disregards Macduff as a threat.
In act 5, scene 4, Malcolm and his army are at Birnam Wood and decide to use branches from the trees to conceal their numbers as they approach Macbeth’s castle. Also in act 5, the reader learns that Macduff was born via caesarean section (“from his mother’s womb / Untimely ripped”). Both revelations show that Macbeth should have carefully considered the possibility of the prophecies coming true.
Ultimately, Macbeth is killed on the battlefield by an inferior fighter—just like Achilles by Paris—because he did not entertain the possibility of defeat.
Achilles and Macbeth were warned of the circumstances surrounding their deaths, but neither heeded those warnings. Therefore, their pride costs them their lives.