Alexander conquered the Persian Achaemenid Empire so quickly because he kept going straight for its king. Just four years after he crossed into Asia, King Darius III was dead and the empire's rule had collapsed.
A king at twenty
Alexander III of Macedon became king in 336 BC, at the age of twenty, after his father Philip II was assassinated. Within about a decade he would lead his army from Macedonia deep into Asia.
Crossing into Asia
In May 334 BC, Alexander crossed into Asia and fought the Battle of the Granicus. It was the first of three major battles between Alexander and the Persian Achaemenid Empire.
Two battles against Darius
On 5 November 333 BC, Alexander's Hellenic League met the army of Darius III at the Battle of Issus in southern Anatolia. It was the second major battle of his invasion, and he beat the Persian king himself.
Two years later, in 331 BC, the two kings met again at the Battle of Gaugamela. It was the second and final battle between Alexander and Darius III. After it, Alexander marched into Babylon.
From Persepolis to the Hydaspes
Persepolis was the ceremonial capital of the Persian kings, and by 330 BC it was in Alexander's hands. That same year, Darius III, the last Achaemenid king, died.
Alexander then continued east. In May 326 BC, he fought the Battle of the Hydaspes on the banks of the Hydaspes River against Porus, far from where he had started. He died in 323 BC.
The short answer
He kept striking at the Persian king, and four years after crossing into Asia, Darius was dead.