This question came up on Quora earlier today and I figured it was worth answering. Although there is no way to estimate Alexander's IQ with any degree of confidence (he obviously never took an IQ test or anything remotely similar to one), we do have plenty of clues as to his intelligence. Here are some things to keep in mind:
Alexander the Great's personality is shrouded in mystery. The best descriptions we have were written hundreds of years after Alexander's death.
Alexander the Great was literate and, by some accounts, an avid reader. He requested volumes of famous Greek plays and literature be delivered to him while campaigning in Asia. His favorite book was the Iliad. His tutor, the great philosopher Aristotle, gave Alexander an annotated copy of the epic poem, which Alexander kept with him throughout his travels.
Alexander was an incredibly skilled general. He defeated much larger armies on their home turf multiple times. He never lost a battle (although he did experience multiple setbacks - lookup his siege of Tyre and his army's struggles in Bactria). Alexander's ability to think on his feet in the heat of battle could be seen as a sign of high intelligence.
Despite Alexander's tremendous battlefield acumen, he was not responsible for turning the army of Macedon into Greece's strongest fighting force. Most of the credit for Macedon's rise into the Greek hegemon belongs to Alexander's father, Philip II. However, Alexander should get credit for leveraging Macedon's progress into the world's largest empire.
Alexander the Great was almost certainly a very intelligent person, but there is no reason to think he was a genius in the academic sense of the word. His most valuable assets were his ambition, his fearlessness in battle, and his ability to motivate his army.