Alexender the Great conquers the Persian Empire and most of the known world, and comes to the land of Israel at about 325 B.C.E.
He meets Simon the righteous, and is so impressed that he spares Jerusalem and the Temple.
The Jewish people remember Alexander and make a commitment that all the first born male children born during that year would be named after him. This is how the name becomes a common name in Jewish families.
In addition, the Jews said were going to revamp our calendar- were going to date not only from the creation of the world, but also from the time of Alexander. This became known is Minyan Shtarot- the official date used on legal documents. Year 1 was 312 B.C.E. which in Jewish tradition was exactly 1000 years after the Exodus from Egypt.
The Jews are now subservient to the Greeks and Greek culture. Greek language and names enter into Jewish life. For example Afikomen, which means desert is originally Greek.

Alexander dies and leaves no successor- so there is a contest for succession. In the South, Ptolemy becomes the emperor, while Seleucus takes control of the North. Both of them were generals over Alexander the great, but now they are competing with eachother.
Israel is lodged between their territories, and so the Jews need to choose between the two generals. The Jews always favored the Egyptian/Ptolemeic Empire, but the North always tried to control Israel and use it as a buffer against the South.
This brings us pretty much to the story of Chanukah.