Arrian, generally regarded as the best ancient source on Alexander's life, states that Alexander had an enduring desire to imitate Achilles and considered the hero to be his greatest rival [...] Regardless of where one stands on this theory, one must admit that the similarities are striking. 

Both Achilles and Alexander were princes, raised under the care of the world's finest tutors. According to myth, Achilles was trained in a diverse range of fields including medicine and survival by Chiron, an especially wise centaur who taught the most famous Greek heroes. Alexander also received something of a liberal arts education. His father, King Philip, was able to secure the services of the philosopher Aristotle, the protege of Plato now widely revered as a founder of modern political and scientific thought. Aristotle probably imbued the teenage Alexander with a deeper understanding of the Homeric stories he had known since his childhood. 

From the beginning, both Achilles and Alexander showed an uncommon affinity, if not an all-out lust, for combat. While still only young men, they became prolific conquerors who led Greek armies on campaigns against powerful eastern civilizations.

The eerie similarity seen in the early lives of Achilles and Alexander is only intensified by the circumstances surrounding their deaths. Both men died young, shortly after the death of their closest companions, Patroclus (companion to Achilles) and Hephaestion (companion to Alexander). Even the way Alexander mourned Hephaestion seems modeled on Homer's account of Achilles' reaction to the death of Patroclus. Like Achilles, Alexander insisted on holding his friend's body for hours on end, refusing to consume food or drink. For Hephaestion's funeral, Alexander constructed an elaborate pyre and cut his hair short (Arrian 7.14). If these accounts are accurate, it appears Alexander fell back on the example of Achilles during the darkest moments of his life.

Did Alexander the Great Really Idolize Achilles?