The rapid expansion of Alexander's empire, followed by the equally rapid fragmentation and dissonance that resulted from his death were so spectacular that the influence of Hellenic culture was like a morning mist across a lake.

One important aspect of this question is to look at how Greek the Macedonians were: they, like the Epirotes, were considered semi-barbaric cousins and it was only through patience and willpower that these outlying Hellenic peoples were able to gain acceptance within the Greek world. The Greeks, as we commonly refer to them, are romanticized as the noble and democratic Athenians, the militant and unwavering Spartans, and (to a lesser degree) the arrogant and capable Corinthians. These proper Greeks tended to look down their noses at the less storied poleis and it could be said they did think themselves superior to their cousins. But we can't necessarily rule out that the poleis and leagues and kingdoms that were outside greater Greece (which, one might say, is everything south of Thermopylae) were not also Greek. They all had different traditions that might or mightn't mesh well with those of another poleis. Because of this disparity between the traditions in each region, there cannot be any one Greek culture which would act as the be-all-end-all of an over-arching cultural identity.

With that little nitpick out of the way, we can safely say that the waxing of Macedonian power allowed for a convalescence of the Hellenic peoples into a much more coherent coalition. With the success of Alexander's campaign bringing the Hellenic sphere to such great bounds (from Epirus to Pakistan). Alexander did not necessarily preach a practice of cultural genocide; he is noted for wanting to allow the cultures to mesh and meld with his own Greek culture, allowing a more powerful and encompassing culture to evolve. With his death, however, the Diadochi that carved out the successor kingdoms began their own cultural policies.

However, there were not massive deposits of ethnically Greek settling all over Alexander's fractured empire. Portions of the forces lead by the Macedonian lieutenants would, indeed, settle in conquered lands with either their families or with the families they created on the campaign.

Some areas, such as Baktria (in modern day Pakistan/India), had these mixed families as a staple of their population centres (Alexandria Arachosia being an example) in which the native peoples once under the Persian yolk were quick to switch their allegiance to that of the Macedonians. Others, such as Egypt, would not have Greek populations that rivaled those of the natives, resulting in a sort of chasm between the Greek and Nile cultures within the (Ptolemaic) kingdom.

But the influence of Hellenic culture and ideas was not something to be put aside lightly. Considering the fact that the Greek language would perforate the fledgling Roman kingdom and easily come to dominate many aspects of Roman and Italian life, the Successor Kingdoms were easily considered "Greek" in any sense of the word. While their peoples were not ethnically Greek, many would speak Greek as a second or third language (maybe further down the list, but that is neither here nor there), they would fight in the Macedonian fashion (but still act as auxiliaries when fighting in their native fashion), and would spread stylized Hellenic art and crafts throughout their regions.

Eastern cultures were notoriously strong, however, and the Hellenic influence within those Oriental regions typically evolved into a hybrid culture (Alexander got his wish?) that was neither entirely one or the other. Put plainly, the most Greek aspect of the Successor Kingdoms was their ability to adapt and grow within the native cultures without becoming too dominant or too overshadowed.

A good cursory reading on the range of influence Greek culture had in the ancient world is Robin Lane Fox's The Classical World. It is a fantastically fun read, and I'd highly recommend it.