Question: Why Are the Greeks Called Hellenes
Answer: The name of 'Greece' is 'Hellas' (and the adjective 'Greek' is 'Hellenic'), according to the inhabitants of Hellas, who are themselves Hellenes. The name comes from Hellen who was not the woman famed from the Trojan War (Helen of Troy), but the son of Deucalion and Pyrrha of flood fame, who was named Hellen. Hellen, the male, has two ls in his name; whereas Helen has only one. It should be noted that Greek doesn't have an "h," but what is referred to as rough breathing. Hellenistic Greece refers to the period in Greek history following Alexander the Great. Egypt, and particularly Alexandria, came to be the center of Hellenism. The end of the Hellenistic World came when the Romans took over Egypt, in 30 B.C., with the death of Cleopatra.
Read what Thucydides writes about the Hellenes and the unity of the Greeks:
Before the Trojan war there is no indication of any common action in Hellas, nor indeed of the universal prevalence of the name; on the contrary, before the time of Hellen, son of Deucalion, no such appellation existed, but the country went by the names of the different tribes, in particular of the Pelasgian. It was not till Hellen and his sons grew strong in Phthiotis, and were invited as allies into the other cities, that one by one they gradually acquired from the connection the name of Hellenes; though a long time elapsed before that name could fasten itself upon all. The best proof of this is furnished by Homer. Born long after the Trojan War, he nowhere calls all of them by that name, nor indeed any of them except the followers of Achilles from Phthiotis, who were the original Hellenes: in his poems they are called Danaans, Argives, and Achaeans. - Richard Crawley translation of Thucydides Book I