What comparisons between Macbeth and the story of Achilles can be made?

Achilles appears in Homer's epic The Illiad. Achilles has come to be known as one of the most well-known heroes in Greek mythology. As the story goes, Achilles's mother, Thetis, dipped him as an infant into the River Styx. She did this in order to make him invulnerable to all that he would face. Unfortunately for her, the waters of the river did not cover him completely; she left a small place on his heel vulnerable to attack. Achilles ultimately dies from an arrow being shot into his heel, where the water did not touch. This injury proved to be fatal, and Achilles dies.
Similar to Achilles, Macbeth (from William Shakespeare's Macbeth) possesses his own tragic flaw. Macbeth's ambitious nature exists as his tragic flaw. Macbeth's ambition to be king of Scotland proves to be the one thing which means more to him than anything. Because of this, Macbeth is unable to see the unfortunate events which will lead to his own demise.
Another similarity between Achilles and Macbeth lies in prophecy. When Achilles is nine years old, a prophet (seer) tells Thetis that he will die in battle against the Trojans. In order to save her son from this fate, she dresses him as a girl and sends him away. In the end, Achilles decides to join the Greek army, which puts him into war with the Trojans, and he loses his life. Although he is thought to be invulnerable, his mother's actions to save him from the prophecy fail. Achilles's free will seems to be the one thing that ensures the prophecy comes true, resulting in his death.
Macbeth similarly has a prophecy regarding his own death. Three witches tell Macbeth that his death will only come under three circumstances: Macduff will kill him, only a man not born of woman will kill him, and he will not die until Birnam Wood moves against him. Macbeth, like Achilles's mother, tries to use the prophecy's weaknesses to see that the prophecy has no possibility of coming true. Macbeth dismisses the first and second prophecies together. He states that he need not fear Macduff because Macduff is born of woman. He dismisses the final prophecy by stating that trees will not pull their roots out of the ground and walk. Therefore, he need not worry about the prophecies at all. In reality, Macbeth's own free will "allows" him to dismiss the integrity of the prophecies. In the end, Macbeth should have feared Macduff because he was not born of woman; he was born by caesarean section. While the trees themselves do not move, it does look like they do. The soldiers moving against Macbeth's castle are ordered to cut down branches and carry them before them. They do this so that it looks like their numbers are greater than they actually are.
One final, yet arguably less important, comparison between Achilles and Macbeth lies in the idea that both men were well-known and successful warriors. Achilles becomes well-known over the course of the Trojan War's battles. The war has gone on for many years, and time and time again, Achilles has proven himself a great success. Macbeth's renown in war comes as his successes in battle are relayed to the king, Duncan. Macbeth defeated Macdownald, a great foe of the king. Because of the success Macbeth has found on the battlefield, Duncan rewards him with a new title, Thane of Cawdor. The previous thane has been found guilty of treason, and Duncan needs a new thane. Given the news brought to him about Macbeth's successes for Scotland, Duncan rewards him. Essentially, both men are great warriors.