The only texts from Jesus' province that were potentially written during his lifetime are a handful of inscriptions, and a subset of the Dead Sea scrolls. The inscriptions are things like public official dedications, names on statues, etc.; the Dead Sea scrolls are thought to be mostly collections of older texts, not so much freshly written Essene texts.

There is no conceivable reason why anyone might expect public inscriptions to mention a contemporary cult leader -- though if you're interested, there's one that mentions the Roman prefect Pontius Pilate in connection with a temple giving divine honours to the (ex?-)emperor Tiberius -- and no reason to expect the Dead Sea scrolls to mention the leader of a competing sect.

Testimony about ancient individuals and events is very rarely contemporaneous with those events and individuals. When you're looking at figures who aren't generals, senators, or emperors, you get an even more pronounced lag in the literary testimony -- and, in addition, normally you get much less testimony.

I mean, that's unless you're expecting a record of Jesus' death to appear in Seneca the Elder's rhetorical exercises, or Velleius Paterculus' history of Rome -- those are the only literary writers in the Mediterranean world who were writing at the time of Jesus' ministry and whose works survive in anything more than tiny fragments. But of course that would be insane.

[Edit. Though as a curiosity, I now see that one of Seneca's sons, L. Junius Gallio Annaeanus, actually gets mentioned in a New Testament text, at Acts 18.12-17, as a proconsul (ἀνθύπατος). But of course that's got nothing to do with Jesus himself, it's a couple of decades after Jesus' death, and it's in Corinth, not Jerusalem.]

There are some great answers to this question in the FAQ.

The short answer is that we don't have any firsthand accounts of Jesus' life (that is, accounts from actual witnesses), nor accounts that date from his lifetime. That really isn't unusual; we don't have very much information about any other Jewish peasants from that era because historians mostly wrote about the upper classes back then. The earliest texts that mention Jesus are Paul's epistles, usually dated from the 40s or 50s A.D., and the earliest account of Jesus' life is the Gospel of Mark, usually dated no earlier than 60 A.D. and usually between 65 and 70 A.D.

I'm using Bart Ehrman's "The New Testament and other Early Christian Writings" as a source. It's pretty accessible, and is often used as an entry-level college textbook. Would make a great starting point for further reading.

The BBC's 3-part series, The Roman Invasion of Britain, says that Virgil was basically considered the Shakespeare of his day. The series mentioned him in relation to esoteric references to his work on some Roman coins, effectively serving as propaganda.

As a follow up...how much credibilty do historians give Josephus?