Macedons invaded and took over the Greek kingdoms:
Where is the source on this? According to this[0]:
> The surviving public and private inscriptions found in Macedonia indicate that there was no other written language in ancient Macedonia but Ancient Greek
So Macedonia was a kingdom where the aristocracy spoke Attic and later Koine Greek, and the vernacular being either a Greek dialect or other Hellenic language. So the written language would have been Attic/Koine. Alexander, being royalty, would have spoken and written in Attic/Koine Greek.
[0]: https://en.wikipedia.org/wiki/Ancient_Macedonian_language
Sure but... Alexander and his countrymen didn't speak that language or use that alphabet, and were not of that people. You might as well say he was an Ottoman Turk, or a Roman.
To clarify, are you talking about modern Macedonians, or ancient Macedonians? The modern Macedonians speak a language similar to Bulgarian and use the Cyrilic alphabet, both of which were not around in the time of Alexander.