What do the historical methods that we apply to other historical figures in antiquity tell us about Jesus? Scholars disagree on many details, but I will start where most scholars agree before offering some more debated options. (Starting with areas of scholarly agreement is not meant to claim that we scholars are always right, but blog posts must be more concise than books. Fuller documentation appears in my 800-page "Historical Jesus of the Gospels.")
First, the vast majority of specialists, whatever their religious perspectives (e.g., Jewish, agnostic, Christian) agree that Jesus existed. Those who doubt that this is the predominant view have not read widely in the academic literature on the subject. Popular writers sometimes compare Jesus instead with mythical figures who lived centuries before the sources that speak of them. Yet most of what we know about Jesus comes from documents written only a generation after his ministry -- that is, in the same time frame in which the most reliable ancient biographies were written about historical figures. Scholars diverge on how reliable the details in these sources were, but very few accept the idea that Jesus' existence grew up as a myth within one generation. No movement in the Roman Empire had incentive to fabricate its origins in a founder executed for treason. (If one wished to commit suicide, there were simpler ways to do it.)
Most scholars also accept some other claims about Jesus that seem pervasive in the early sources. First, Jesus embraced the socially marginalized. Although at least some of the Gospels address a more mainstream clientele, they preserve reports about Jesus focusing on the sick, the poor, children, as well as groups that polite society considered immoral, such as tax collectors (probably viewed as collaborators with the occupying empire) and those broadly designated "sinners." One might expect someone with political ambition to cultivate instead the favor of elites, yet the Gospels frequently portray Jesus in conflict with groups more respected in his society. They depict his execution as engineered by elite aristocrats and carried out by a corrupt governor -- a pattern that fits the social reality of much of the first-century Mediterranean world.
Second, virtually all extant ancient sources about Jesus, including those authored by critics of his movement, portray him as a healer. Most scholars about Jesus, again across the ideological spectrum, believe that his contemporaries experienced him as a healer. Scholars differ in their understanding of how he healed: some, for example, appeal to the placebo effect or coincidence, others to the curing of psychogenic disorders, and others (especially many Christians) cite supernatural activity. Although the explanations offered differ, the portrait of Jesus as healer fits his reported concern, noted above, for the marginalized and needy.
Third, scholars frequently see Jesus as leading a restoration movement. Our earliest sources, including a letter contemporary with the group, mention a group of 12 special leaders. From the contemporary Dead Sea Scrolls, it appears that a movement working for the renewal and restoration of Israel might indicate this mission in part by having a group of 12 leaders. The first-century Jewish historian Josephus presents both John the Baptist and Jesus in ways that fit this expectation. Many scholars also believe that Jesus' challenge to Jerusalem's elite, epitomized in overturning tables in the temple, fits some contemporary Jewish expectations about the restoration of the temple.
Fourth, Jesus's teaching about God's "kingdom," or "reign," is so pervasive in the earliest sources that most scholars believe that this was a central issue in Jesus's teaching. Scholars often differ as to whether Jesus believed that the kingdom was already present in his ministry, was about to come, or both. In any case, this theme helps explain the urgency of his mission in renewing Israel. Again, however, instead of seeking to establish or invite God's reign in political or military ways, as at least some of his contemporaries did, Jesus focused on the socially powerless. His reported teachings about nonviolence and absolute dependence on the Father cohere with his own behavior in accounts about his death.
My following suggestions are more debated. First, if Jesus led a movement involving the kingdom, he probably expected to play a significant role in that kingdom. After all, the 12 leaders were his own disciples. Nearly all scholars agree that the Roman governor executed Jesus on the charge of sedition, as one who sought to be king; moreover, Jesus's disciples also came to believe that he was a king (the Messiah). No circles of early Christianity known to us explicitly repudiated that idea. Did the governor get the idea from Jesus' disciples? Did they get the idea from the governor? Or do the dots connect through Jesus, perhaps late in his public ministry? Interestingly, the early 20th-century scholar who popularized the view that Jesus did not believe himself Messiah later reconsidered it.
Second, if Jesus publicly challenged the elite, spoke about a present or imminent kingdom, and yet refused to resort to violence, he probably expected to die. Many scholars have doubted that Jesus predicted his death, suggesting that it was an unexpected tragedy. Perhaps a popular leader from mostly rural Galilee failed to anticipate the kind of political power he would be facing in Jerusalem. Yet Judaism already had a strong tradition of martyrs. Further, Galileans regularly traveled to Jerusalem for festivals, and the power of the elite there was well known. Indeed, many diverse Jewish voices from the period -- Pharisees, the Dead Sea Scrolls, Josephus and others -- criticize some members of the elite priestly families for alleged abuse of power.
Jesus' reported debates and especially his reported disruption of temple activity by overturning tables would challenge the elite's honor, and in the empire, local elites were responsible to detain troublemakers. If Jesus' preaching of an imminent kingdom sounded too political, it would arouse Roman concern. Jesus probably did not simply predict his death: He provoked it. If he expected to fulfill a role in the kingdom, and if, like many of his Jewish contemporaries, he expected a future resurrection of the dead, Jesus may have extended his dependence on the Father to trusting that the Father would raise him from the dead. Unlike some other points, whether such an expectation was or ever will be fulfilled is a matter on which scholars of divergent religious convictions will not as easily agree.