It is important to remember that the Greek world was a collection of separate city-states as opposed to its modern form. Each city had its own government, army, navy and other such institutions, and as a result there was never a true Greek Empire. Though they did occasionally unite (such as to fight the Persians), mostly they were at war with each other and using countries such as Macedonia as allies. Sparta built up one of the longest empires in the Greek world, attaining control of the Peloponnesian peninsula.
The Athenian Empire was created via unification of individual Greek city-states and micro-states for the common purpose of defending Greece and driving the Persians out of Greeks. Founded 478 BC and dissolved in 404 BC, at the end of the Peloponnesian Wars due largely to poor organisation and leadership.
The Macedonian Empire (not to be confused with the modern nation that was renamed from Vardarska to Macedonia by Tito at the end of the second world war), which is also the empire that is commonly known as the Greek Empire or the Hellenic Empire was created in 808 BC largely by unification for mutual benefit at the north coast of the Aegean Sea, and was conquered by the Roman Empire in 146 BC.
The Byzantine Empire, which was known as both the Second Roman Empire and the Second Greek Empire (the first referring to the Macedonian Empire), was originally known as the Eastern Roman Empire, as the Roman Empire had become too large to manage and was split off into two halves. Rome was split off into the two halves in 286 AD. The Eastern Roman Empire is thought to have been largely Hellenised by the time the empire was split, and was pretty much officially Hellenised by 620 AD, under the rule of Heraclius (in power from 610 AD to 641 AD) when the language of the now Byzantine Empire was changed from Latin to Greek. The Byzantine Empire reached it's peak under the Macedonian Dynasty (roughly 800 - 1000 AD) (note again, Macedonian as in Greek Macedonian, not as in Vardarskan), which also happened to be a time of major renaissance of the Greek Culture within the Byzantine Empire. The Byzantine Empire fell in the famous year of 1453, when the Ottoman Turks (I specify Ottoman as there were other Turks such as the Seljuk Turks, Uzbek Turks, and Huns) brought down the walls of Constantinople with the never-before-seen Abus Guns, which were essentially some of the first Cannons in Europe/Anatolia.