Would the historical Jesus have been a "Romanized or Hellenized" Jew? As I understand it, Jesus lived in an era where Judea had become very culturally influenced by the Greco-Roman world. Would this extend to fashion and style? How Hellenized was the world he lived in?

First, I'm away from my books at the moment (redditing while on my night shift!). So I need to just use the eBooks I have on hand.

It needs to be mentioned that the Hasmonean dynasty is sandwiched between Macedonian and Roman rule; while attempts to Hellenize the Temple are cited as the spark to revolution in 1 Maccabees, Rome is spoken of positively in 1 and 2 Maccabees (Rome offers some form of support, off-hand I cannot recall if the support was symbolic or material; regardless, Roman power is not portrayed antagonistically). In Jesus' time, Hellenism had been the cultural norm for three centuries. Hellenism, Thomas Yoder Neufeld writes, "would constitute the world culture well into the Roman period, allowing for an unprecedented level of commerce in goods and idea, facilitated by a common language, which many people could speak in addition to their local tongues." Under Herod there were deliberate attempts to Hellenize, evidenced through construction projects and the fact that Herod's children were educated in Rome. Though such overt Hellenization likely did not reach the average rural inhabitant. But Nazareth was located close to Gentile urban centres; if Jesus' family were day labourers treks into these cities would not have been unheard of.

The Gospel of Mark gives a clear impression that Jesus spoke Aramaic, but if we consider the accounts of Jesus (and, later, Jesus' followers) interacting with Gentiles we could probably include that they were speaking some sort of common language -- and Greek seems to be the likely candidate if non-Jews and Jews were speaking to each other.

Stanley Porter supports this conclusion:

Recent research has widely accepted the notion that Judaism out of Palestine was Hellenistic, or perhaps better Hellenistic-Roman, because of the obvious fact that, even if Jews attempted to live in community within the larger Greco-Roman world, they could not help but be in the midst of and interact with such a world.... Admittedly Hellenism was not a homogenous cultural phenomenon.... Jews, like others in the Mediterranean world of the time, used Greek as their lingua franca.... The question of the Hellenism of Palestine has been a more highly contest issue.... The New Testament records incidents where it is likely that Greek was spoken by Jesus and his followers (e.g. Pilate in Mark 15:1-5 and parallels; the Syro-Phoenician "Greek" woman in Mark 7:24-30), and Acts 6:1, 9 bear witness to Greek speaking Jews and a synagogue in Jerusalem.

Yoder Neufeld writes that Galilee had a "mixed population of Jews and Gentiles" but notes that there's not a scholarly consensus on to what degree these populations might have interacted. But Sepphoris, a Greek urban centre, was a few kilometers away from Nazareth. And "[a]long the shores of the lake where Jesus appears to have been active was Tiberia, the imposing new city named after Roman Emperor Tiberius" which had been built as a way to for the Herodians to honor their Roman benefactors. Yoder Neufeld supposes that Jesus might have avoided these cities, but his basis for this argument is weak (he suggests the imagery of the parables suggests that Jesus primary worked in rural areas; I like Yoder Neufeld but this is just a weak argument).

Porter helpfully points out that it's really better to think of the context as both Jewish and Hellenistic, rather than one or the other. Jew and Gentile is the common dichotomy we are given in the Bible, but it needs to be stated that this simple dichotomy is not always taken to name reality so neatly. I think it's hard to say that Jesus and His followers would not have been somewhat Hellenized. This has tended to be a contentious issue because it's been viewed as a binary -- Jesus is either Hellenized or Jewish -- when the reality may very well be that some degree of Hellenism might have been a given. This does not mean that Jesus was fully Jewish, but rather that the character of that Jewishness might have been informed by the cultural context Jesus found Himself in.

Sources:

Stanley E. Porter, "The Context of Jesus: Jewish and/or Hellenistic," Vol. 2 of Handbook for the Study of the Historical Jesus, eds. Stanley E. Porter and Tom Holmén, Leiden: Brill, 2011. eBook accessed via EBSCOhost

Thomas Yoder Neufeld, "Jesus' World," from Recovering Jesus: The Witness of the New Testament, Grand Rapids, MI: Brazos Press, 2007. Kindle edition.