Almost everything I’ve read about Judaism in the time of Christ pertains to Judean Judaism. But some remarkable things had been happening in and around Alexandria. There was a large Greek – speaking (“Hellenistic”) Jewish community. They used the Septuagint, and were familiar with Greek philosophy. While the Septuagint in any English translation differs only a little from the Hebrew text, the Greek translation used a number of keywords with strong connections to secular Greek philosophy, including “En arche” (“In the beginning”, Genesis 1:1; also in John 1:1); “Logos” (“Word” – indirectly, as in “God Said”, in Genesis 1:3); and ” ‘o On” (“The One who is” – “on” being the root of our word “ontology”, the study of being) (in Exodus 3:14).

In John’s Gospel (7:35), when Jesus says that he will go away for a little while, some of the crowd wonder whether He will go to the Diaspora (Greek / English interlinears use that exact word – English translations vary) and teach the Greeks.

It was natural for those Greek – speaking Jews to look for common ground between Hebrew scripture and secular (I prefer that to “pagan”) Greek philosophy. And Jewish scholars like Philo found a great deal of common ground, noting the parallels between the philosphers’ “Logos doctrine” and the Hebrew “Wisdom” tradition (Proverbs 8:22 et seq). Modern Jewish author Daniel Boyarin, in his book “Border Lines”, makes a case for the existence of a substantial community of Hellenistic Jews who saw the “Logos” as a person – to some, a “second power in heaven”. This brought them right to the brink of what Boyarin called “binitarianism” – the belief that God might have begotten a son.

These developments, if I’ve stated them properly, have profound consequences for both Judaism and Christianity.
They lead me to make a number of observations and speculations, on which comments will be greatly appreciated.

The Jews of Jerusalem (and Judea in general) were looking for a Messiah, but not for a Son of God. I don’t know whether the Temple authorities had gotten word of what was going on in Alexandria, but if they had – being, presumably, firm believers in the Deuteronomic view that the Lord is one – they might have been prejudiced against Jesus before He even began His ministry.
The Jews of Alexandria (and in other centers of the Diaspora, including Ephesus, where John moved his community when the environment in Judea became too hostile) would seem to be open to hearing about a Son of God, but perhaps not so much about a Messiah. These differences might account for the differences between John’s Gospel and the Synoptics about who Jesus was, and why he came here.
The prologue to John’s Gospel (up to verse 13) has much in common with Philo’s writings. Robert M. Grant, in “Jesus After The Gospels”, sees a great deal of literary common ground, although not enough to draw a firm conclusion. In John’s first epistle, the first verse of Chapter 2 (“If any man sin, we have an advocate with the Father…”) comes pretty much directly from Philo. Of course, John then added the words which only he could have supplied – in verse 14 of his Gospel, “And the Word became flesh, and dwelt among us”, and in verse 2 of his Epistle, “Jesus Christ the righteous.”
My general feeling about John’s writings is that they are closely tied to the Septuagint. I suspect that Jerome might have valued the parallel construction between John and Genesis that when he developed his Vulgate, he began both books with “In principio” – one of the few times when he chose the Greek text over the corresponding Hebrew.

In the second century, Aquila revised the Septuagint. One of his many changes was to remove “En arche” from Genesis, and replace it with a word (which I haven’t been able to locate) derived from Hebrew. Some observers have said that he did this to make the text less useful to Christians. But I believe it is more likely that he did this to break the connection between the LXX and Greek secular philosophy. 
I’ll conclude with possibly my most speculative line. In order for Jesus to carry out his mission on Earth, he needed one group of people who would hand him over to be crucified for claiming to be the Son of God, and another group of people who would build a Church around His life story. And that’s precisely what he got.