It's true that he was Macedonian and that he is a point of debate for the legitimacy of different nation-states in the Balkans to contest over (but let's not get into that). The OCD lists the Macedonian Kingdom under Philip II as including

Northern barbarians... Greek-speaking Upper Macedonians... Illyrii, Paeonians, and Thracians - with their own non-Greek languages - and Calcidians and Bottiaeans, both predominantly Greek-speaking.

About the Macedonians themselves, Brill's New Pauly says

At the time of the Macedonian conquests, the population was of different origins, but Hes. fr. 5 already considered them a Greek tribe of horse lovers, descendants of Zeus, residing in the areas around Pieria and Olympus. The first royal dynasty, the Argeadae, claimed descent from the Temenidae of Argos [II 1] in the Peloponnese. Thus, Alexander [2] was able to convince the judges in Olympia of his Greek origin (Hdt. 5,22). If in later disputes, especially during the controversies with Athens in the 4th cent., some of the southern Greeks maintained the Macedonians - with the exception of the royal house - were not Greek at all (cf. Isocr. Or. 5,107), the argument was not only based on current political interest. Rather, it reflects the historically somewhat obscure genesis of the Macedonian state, as well as the different political and social structures when compared with the póleis. The latter mostly could point to a much longer political history, and the Athenians even claimed to be autochthonous, i.e. they had always been there. No Greek state of the period had an all-powerful royalty, and the Macedonian dialect (Macedonian), while probably being basically Greek, was intermingled with Illyrian and Thracian words and phrases and therefore was considered unintelligible, therefore non-Greek [1; 2]. [Emphasis mine]

Philip was actively hellenizing at least the Macedonian court, importing for instance Aristotle to tutor Alexander, hosting Euripides, becoming a member of the Amphictyonic League, among other things. Here's a map (granted, from Wikipedia but it makes the point) of the extent of Macedonian influence under Philip II. Note the location of Pella, Macedonia's capital, just under the mark "Macedonia" on the map.

The question of whether Macedonians were Greek was an active one used in the ancient world for purposes of political legitimation. Philip clearly felt the need to prove his Greek chops by cultivating a Greek literary and intellectual culture at Pella, but at the same time included in his territories many non-Greek speaking people. Alexander was Greek speaking, Greek educated, probably (judging from his reported admiration of Homer) was probably an admirer of Greek culture. His father's kingdom included a number of non-Greeks as well as Greeks, and Macedonian ethnicity was debated during the late Classical period... so go figure?

Alexander's mother, Olympias, was Philip's 4th wife, which was arranged due to political implications. Philip used diplomatic marriages to secure his borders (among other tactics) but the union with Olympias was an example of this.

Olympias was the daughter of King Neoptolemus I of Epirus, which is the modern day state of Albania. This thread has some great examples of arguing whether the Macedonians were perceived as "Greek" at the time, however in the literal sense, yes, Alexander has half-Epirus descent, in what is today Albania. Now with the centuries of cultural diffusion, it is too bold to say that he is Albanian, but his family would have come from the region at that time