For about a century or century and a half there is a cold war (sometimes a hot war) between the Northern and Southern Kingdom. Israel is caught in the middle.
The influence of Greek language, culture, etc. comes strongly into Jewish life. However, there are many things in Greek culture which are antithetical to Judaism: nudity, Paganism, obsession of the body, etc.
In the Ptolemaic empire, Ptolemy gathered 70 Jewish scholars and ordered them to translate the bible into Greek. The Rabbis say that there was an aura of divinity about this, since each of the 70 were in different rooms and by all logic should have had slight variations between the translations- however, they all translated it in exactly the same way. This translation is called the Septuagint.
This is the famous translation of the Bible that eventually came into Latin and English, eventually the Bible that the world knows.
The Rabbis were of a mixed mind, they on one hand regretted translating it into Greek- on the other hand they supported it.
Every translation has differences from the original, there are slight differences in nuance- especially from Hebrew which has a very small vocabulary and therefore much of the meaning is conveyed by nuance and slight letter variations.
These small differences become exaggerated in translations from the translations. For example Hebrew to Greek to Latin to English, some things become problematic.
For example- one thing the Jews have always contested is that the Catholic Church has a belief of a virgin birth based on text in Isiaiah. The original text uses the Hebrew Almah which means a young woman- not a virgin. A virgin in Hebrew is a Betulah. However, Almah when translated into Greek can mean either young woman or virgin, when translated into Latin it became virgin, and the mistranslation crept into the English translation even till today.
The Greeks had both negative aspects of their culture and positive. Many of these became contentious points between the Greeks and the Jews, but overall as long as the Greeks let the Jews live their lives, there were no problems and the Rabbis spoke highly of the beautiful parts of Greek culture.
Once the Greeks attempted to force the Jews to become Greek- that became a different story. By the Second century, the Northern empire controls the land of Israel, and so the clash between the two becomes enevitable. The Greeks insisted that Jews eat pork, abstain from circumcision, and put an idol of Zeus in the Jewish Temple in Jerusalem. Out of this comes a great Jewish resentment to the Greek culture and to the hands of the Greeks which rule them. Out of this comes the story and holiday of Chanukah which is celebrated till this very day.