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

If we look first at Macbeth and Achilles, we can say that they are both great warriors. Because he is the son of a goddess (Thetis) and was dipped in the River Styx, Achilles is virtually unbeatable in battle. As Macbeth opens, we learn of Macbeth's great feats in battle:

For brave Macbeth—well he deserves that name—
Disdaining fortune, with his brandished steel,
Which smoked with bloody execution,
Like valor’s minion carved out his passage
Till he faced the slave;
Which ne'er shook hands, nor bade farewell to him,
Till he unseamed him from the nave to th' chops,
And fixed his head upon our battlements.

In other words, Macbeth courageously killed many men to defend Duncan's throne, including one he chopped in half.
Both Macbeth and Achilles kill a royal. Macbeth kills King Duncan, and Achilles kills Hector, a prince who is heir to the Trojan throne. Both Macbeth and Achilles also behave dishonorably. Macbeth murders Duncan when Duncan is a guest at Macbeth's home, which violates all the rules of hospitality in his society. Macbeth is supposed to do everything he can to protect his guests, not kill them. Likewise, Achilles dishonors Hector by dragging his body around behind his chariot after he has killed him (though he does later allow a funeral for Hector).
Both men cause damage to their countries through their own weaknesses. Macbeth's weakness is his ambition, and Achilles's weakness is his pride and anger. Macbeth ends up bringing misery to Scotland by being overly ambitious, which leads him to become a tyrant. Malcolm, the rightful heir, brings an invading army to the country to win back the throne, plunging Scotland into war. Achilles almost deserts his countrymen, leaving them in danger of losing the Trojan War, because of his pride and rage.
Because Achilles will not fight, his best friend Patroclus dons his armor and is killed in battle; it is because of Achilles's pride and rage that his friend dies. Likewise, Macbeth has his good friend Banquo murdered because of his ambition: he is afraid Banquo knows too much and will cause him to lose the throne.
Stepping back a bit from the two main characters, it is notable that the story of Macbeth and the story of Achilles are both influenced by a supernatural element. A prophecy from the three witches, who are supernatural creatures, leads Macbeth to his bloody path, though he always had a choice to take a different path. A supernatural prophecy also surrounds Achilles, and it offers him two choice in life. He will either have glory and a short life or no glory and a long life:

For my mother Thetis the goddess of the silver feet tells me
I carry two sorts of destiny toward the day my death.
Either, if I stay here and fight beside the city of Trojans,
my return home is gone, but my glory shall be everlasting;
but if I return home to the beloved land of my fathers,
the excellence of my glory is gone, but there will be a long life
left for me, and my end in death will not come to me quickly.

Both Macbeth and the story of Achilles are exciting, violent tales with supernatural elements (the gods, such as Zeus, are always getting involved with Achilles, as the witches do with Macbeth). Each story examines the character flaws of two brave warriors, though Macbeth's flaw are darker and more damaging.