The truth is Alexander achieved so many things, even at a point he was called 'Alexander the Great.' I think Alexander's biggest accomplishment is that he was able to fulfill his destiny. He grew up knowing himself as a predestined child, and he was able to prove it.
He demonstrated this by fulfilling his father's mission. All his father had wanted was for the whole of Greece to be under him so that the same culture and idea could spread throughout every nook and cranny of Greece. Alexander the Great inherited his father's seat after his demise, and he was able to conquer almost all the territories around him.
One of the most important moments of his life was when he fought and conquered the empire of Persia. This really helped him to extend his territory, and he even went as far as launching a campaign to conquer India in 326 BC.
The most significant accomplishment for Alexander was that he was able to conquer more territory than any other general before him. He is also credited as having never lost a battle. This has been attributed to the battle tactics that he employed, which were far advanced than his enemies.
He was a brilliant tactician. Before he became king of Macedonia, as a young man of 16, he was putting down uprisings. But after he became king, he dealt with those who were rebellious close to home, and after building a large army, set out to take possession of the lands around him.
He crossed into what is now Asia and conquered the Persian empire. He marched his men through over 17,000 km, taking one nation after another. No other king or general had managed to do as much as Alexander, especially in such a short time. He reigned over his empire for 12 years, before he died under mysterious circumstances. After his death, the four generals underneath him divided the territories amongst each other.