Good (and somewhat complicated) question.

A fairly wide consensus has emerged that many elements of the gospels cohere with those of bioi ("lives"). The basic structure of a bios might examine an individual's "ancestry, birth, boyhood and education, great deeds, virtues, and death and consequences."

However,

If the canonical Gospel writers were, indeed, consciously following the Greco-Roman bios literary form, they were also creative in their transformation of that literary form. If the bios form served as the literary ‘starting point’, the NT authors used and molded a known literary form to most effectively communicate to their respective audiences. The authors changed the very purpose and impact of the familiar bios genre, because they had a very different purpose in relating the story of Jesus.

But we needn't explain the variance with traditional bioi based solely on the authors' communicative/rhetorical strategies. We also have to remember that the gospel authors - as Jews (or para-Jews) writing in Greek - were heavily steeped in the Hellenistic Jewish literature of the times. They would have encountered the "Old Testament" through its Greek translation (the Septuagint), as well as other Hellenistic Jewish literature in Greek. So it's fairly certain that the gospels were also indebted to some of these literary forms as well, in some ways.

To take a few examples: a fairly recent monograph on the Gospel of Mark focuses on a close connection with the Jewish novels of the Hellenistic period - some of which constitute what we now call the "deuterocanon," usually not accepted by Protestants. As for literary connections with the actual (universally) "canonical" Bible itself: the portrait of Jesus' early life in various gospels syncs up pretty closely with that of Moses (and events from Exodus) - as well as with Elijah, at certain points.

Sources:

• Richard Burridge, What Are the Gospels? A Comparison with Graeco-Roman Biography

• Michael Vines, The Problem of Markan Genre: The Gospel of Mark and the Jewish Novel

• Judith A. Diehl, "What is a 'Gospel'? Recent Studies in the Gospel Genre," Currents in Biblical Research 9 (2011): 171-199

• Michael Martin, "Progymnastic Topic Lists: A Compositional Template for Luke and Other Bioi?," New Testament Studies 54 (2008): 18-41

My background is in philosophical theology, so I'm not the most qualified person in the world to answer this question, but since no one else is taking a swing at it...

IIRC, the exact lineaments of the gospel genre are somewhat in dispute. What's a bit better understood, I think, is all the various genres that make up the subunits of the gospels.

The gospels were almost definitely not written any sooner than the late sixties of the first century CE. But they contain material that seems to go back to Jesus himself. The gospel writers took various bits of tradition that had been handed down by Christians since the beginning and imaginatively stitched them together into their own theological vision. But these little bits of tradition themselves come in different genres. So, for instance, we're all familiar with the genre of the parable, but not everything in the gospels is parable. For instance, Mark 13 is definitely in the genre of apocalyptic - writings which use dramatic endtimes prophecies to deal with current issues. (Basically a powerless minority proclaiming hope that God is about to dramatically intervene/change/end history on their behalf.) You also get miracle stories, allegories, moral exhortation, etc. So, whatever the form of the genre "gospel" turns out to be, the content is a hodgepodge of other little genre pieces.

I know that the Book of Enoch gets directly quoted elsewhere in the New Testament, but I don't know of any references to them in the gospels themselves.

The Gospel of John bears the marks of some kind of influence by Platonic philosophy, though it's hard to pinpoint a particular text. I did hear an argument once that John is referencing Plato's Symposium in the account of the woman at the well, but I got the impression from the scholar making the argument that he himself took it to be a little idiosyncratic.

Sorry I can't give you more or provide good references. I wouldn't break the sub rules so brazenly if it weren't that you've gotten no answers yet! Maybe this will entice some more knowledgable folks to come correct me...

Each gospel shows different characteristics of Hellenistic biography (βίος ), Graeco-Roman history, and allegorical fiction. Luke and Acts purport to be histories, but in a manner similar to contemporary historical fiction. In short, the gospels (canonical and non) represent their own genre apparently invented by the author of Mark, to couch wisdom sayings in a theodicy influenced by Jewish novels like Tobit, Judith, the Greek Daniel and Esther...

There are a number of good, not-too-technical articles on these questions and the history of scholarship on them here: http://vridar.org/2014/01/17/genre-of-gospels-and-acts-index/