The references for this post come from Scythopolis - a Greek City in Eretz Israel written by Gideon Fuks and published by Yad Izhak Ben-Zvi in 1983. Yad Izhak Ben-Zvi is one of the most respectable academic publishing houses in Israel.
The book is somewhat dated, since more information has come to light and old assumptions have been reconsidered, but this is the most concise, accessable work I have a copy of. It is more than enough to show how the world of ancient Judah was far more polyglot than is popularly thought.
The author states that certain aspects of the Greek way of life were known to the inhabitants of Coele-Syria before Alexander the Great conquered the region in 332 BC. Fuks uses Coele-Syria loosely, to refer to everything from Damascus to Judea, inclusive.
He mentions Yigal Yadin's theory that some of the ostraca (pottery shards used for writing) from Arad, a biblical fortress in the south of Judah (which fortress at one point even had a temple), ostraca dating to the 7th century BC, are Greek.
Several ostraca from the same site in the 6th century refer to Chithtites, in all probability Aegean mercenaries employed by King Josiah. At least that is the explanation offered by the archaeologist Yohanan Aharoni.
He also considered the large amount of Greek pottery shards in Mesad Heshavyahu (a fort just north of Ashdod/Azotus) indicative of a further mercenary garrison employed by Josiah.
Several sites have revealed Greek pottery from as early as the 9th c. BC.
From the 7th c. onwards the amount increases.
The author notes that these finds are especially prevalent in layers from the middle of the 6th c. From about the same time come a few Greek coins as well.
The eminent archaeologist and scholar William F. Albright concluded that there were several minor Greek settlements along the coast of Syria, Lebanon and Israel. It should be pointed out that the evidence isn't conclusive (rarely is in archaeology).
The author concludes that although it is hard to measure the Greek influence on the region, it certainly was aware of the Greeks, and Alexander's conquest did not entail a complete culture shock. Of course, the change Hellenism wrought upon all aspects can hardly be overstated, but Greek influence had been slowly creeping in for centuries.
To the above can also be added the presence of the Philistines and other Aegean peoples, as well as the Phoenicians, who had close contacts with the Greeks.