Almost nothing. Weirdly, the best way to get at this might be through Paul, with some historical background to fill things out (it's weird because Paul's surviving writings say almost nothing about the pre-crucified Jesus).

One principle of studying ancient history is that when something can't actively be confirmed (usually by archaeology or other science, or by multiple unrelated reports), the next best thing to confirmation is having no reason not to believe it. That's more or less where we are with Paul. There's no good reason to think at least the core seven Pauline epistles aren't basically authentic--they read like real letters instead of treatises constructed like letters, they were almost certainly written by the same person, based on things like writing style, and they're consistently part of the canonical tradition from about as early as we can talk about anything like a canon. And there's no good reason to think that Paul is lying in them. He's definitely got an agenda and I would argue that he's not always correct in the ways that he represents the communities he's writing to, but there's no reason to think he's just making stuff up.

There are three pretty concrete things we can get from Paul: 1.) the use of the title Christ, which is a Greek translation of the Hebrew "Messiah," 2.) the tradition that Jesus preached against divorce (1 Cor 7, compare Mark 10:2-9), and 3.) that Jesus had a brother named Jacob (this gets translated to James in English, for interesting linguistic reasons that you should Google), who was based in Jerusalem after the crucifixion (Gal 1:19). Other traditions preserved in the Gospels give Jesus a few other brothers, some sisters, a mother, and a carpenter father (see esp. Mark 6), but the Gospels are all after Paul's writings and we know a lot less about their authorship and where their information is coming from. I would say that the family details ring true mainly because they don't have any strong symbolic or literary correlations.

So all of this gives us a picture of a laborer from Judea or one of the nearby areas--traditionally Galilee, and that seems correct for the same reason that the family stuff does. Jerusalem would be symbolic. Bethlehem in Matthew and Luke is symbolic. Nazareth and Capernaum are not symbolic.

Besides that, there's a fair amount of work on the contexts of Judean and Galilean laborers in the first century in general, but the only other thing we have to go on for Jesus himself is the use of "Messiah/Christ" as a title. I'm trying to think of a modern analogue to the political statement that title makes in the first century, and I'm coming up short. Maybe it's like walking into the British House of Lords and declaring yourself King Arthur, except that in this analogy England lost the Cold War primarily by attrition and political maneuvers and is on the brink of civil war. Josephus and the book of Acts tell us about several different claimants to the messianic title, all of whom led or attempted to lead revolts against Roman power. Some scholars try to distance Jesus from that, but it's really difficult to escape if you put aside Christian theology in favor of the historical context. The fact that Jesus is supposed to have been crucified, a highly public and humiliating death often used for rebels, bears this out. The fact that his followers were apparently not executed along with him suggests that he wasn't actively planning violent action (yet??), or at least that Peter et al. knew how to go to ground for a few years.

That's it. People who study the historical Jesus try to fill in more, and there are some good guesses we can make by applying knowledge of first century economics, sociology, etc., to the early traditions, but since there's no real consensus on which parts, if any, of the Gospels are historical, the field of historical Jesus studies as a whole tends to have pretty wide-ranging disagreements.

Consider this answer by /u/talondearg from the FAQ, which discusses what documentation of Jesus exists.

Follow up question building off of the OP: what are good reads that discusses the historical context of the Biblical accounts from a secular perspective? I was raised by pretty hard core Christians (though I'm not religious any more), and I would like to expand my horizons. Unfortunately, I don't really know where to start.