Alexander the Great was both a Greek and a Macedonian. He considered himself Greek, spoke Greek, was in love with Greek culture, and spread it across the world. He simultaneously considered himself Macedonian, however. He was most definitely not Slavic.

Macedonians were a people living to the immediate north of Classical Greece in the area now mostly contained in the modern area known as Greek Macedonia (of which the capitol is Thessaloniki). Ancient Macedonians spoke a northern variety of Greek, as well as Koine Greek once it spread across the world. The modern nation of Slavic Macedonia contains very little of Ancient Macedonia's territory, and actually mostly coincides with an ancient nation called Paeonia.

Slavs didn't actually appear in the Balkans until almost a thousand years after Alexander's death. There's currently a huge political dispute about the naming of The Former Yugoslav Republic of Macedonia. I'm not completely clear on why the nation chooses to call itself Macedonia, but I believe it goes back a few hundred years at least. It certainly creates a lot of confusion, not to mention bad blood with the Greeks in Greek Macedonia.

The slavs migrated around the 6th century to the balkans, that's 1000 years after Alexander. So you can say that Alexander was a Greek, but...it's more complicated. This is part of the complicated nature of ethnicity at the edge of the greek world, a problem that the greek nationalist refuse to consider, fearing (wrongly) that the modern macedonian might steal their heritage. So to come back to Alexander he spoke Greek, he was culturally greek, so in that sense he was greek. But it had two signification, a cultural one, that fit perfectly to ancient macedonian as well as for an hellenized member of the Babylonian elite in the IInd century BC, they were greeks because they wore greek cloth and spoke greek. But their is another sense, more narrow and with a more geographical sense to limit the qualification of greek to the inhabitants of Greece (and the colonies of the greek cities). The ancient geographical limit of Greece was quite narrow as the Thermopylae was already seen as a border. The most important element was that the in the 6th century BC, a king of Macedonia (Alexander Ist) had to show that his ancestors came from Argos in order to prove that he had the right to participate to the olympic games. So at that time the Macedonians were not automatically accepted as greeks.