When Alexander the Great conquered the Persian Empire how did Greek culture get spread throughout it?

Alexander consciously set out to Hellenise it. He established a hundred cities in the Greek tradition (most named Alexandria) as centres of Greek-style culture, with gymnasia, temples, architecture etc and citizenship rules. He married his generals to Persian women and brought the Persian aristocracy into his armies. Greek became the lingua franca of commerce.

This provided a veneer of Greek civilisation which lasted until the collapse of the Hellenistic kingdoms established by his successors and the encroachment of Islamic culture.