Conquest. Alexander the Great spread Greek Culture throughout the world by conquest.
Alexander's armies carried Greek Culture with them everywhere they went. When Alexander was died he had conquered most of the known civilized world.
All of Asia Minor which had been the Persian Empire was now dominated by Greek Culture. The Middle East Syria, Iran, Israel were dominated by Greek CultureCo. ( Causing a rebellion in Israel)
Egypt was part of the Greek Empire Cleopatra queen of Egypt was more Greek than the Egyptian of the pharaohs.
Oddly the conquest of Greece by Rome also spread Greek culture.
The Conquering Romans carried Greek culture back to Rome. Every wealthy Roman family had a Greek slave as a tutor for their children.