Alexander the Great was the Greek Hellenic King of Macedon.
For when Alexandros(*) chose to compete in the games and entered the list to do so, his Greek competitors tried to exclude him by claiming that the contest was for Greek contestants only, and barbarians were not allowed to participate. But when Alexandros proved that his was of Argive ancestry, he was judged to be a Hellene, and proceed to compete in a footrace, in which he tied in the first place.
(5.22, The Landamark Herodotus, The Histories, edited by Robert R. Strassler)
But it only suggests the Greek's ancestry of the Alexandros line. Macedonians as a whole were considered barbarians by Greeks. Additionally they "medized" with Persians, an later intermarried with them. Alexandros I played both sides in the Greek-Persian wars.
(**) Alexandros the Great:
In an attempt to craft a lasting harmony between his Macedonian and Persian subjects, he held a mass marriage of his senior officers to Persian and other noblewomen at Susa, but few of those marriages seem to have lasted much beyond a year.
en.wikipedia.org/wiki/Alexander_I_of_Macedon
(* )Alexandros son of Amyntas (Alexander I of Macedonia c. 498-454 BC)
(**)Alexandros son of Philip II (Alexander III the Great c. 356-323 BC)