Really, the first thing you should do is check out this part of the FAQ, as this is a pretty popular topic on this subreddit. There will be great answers there.

For a quick rundown, the two closest "historians" to Jesus would be Josephus and Tacitus, who both mentioned him. However, they were writing ~ 100 C.E., so not within Jesus' lifetime.

The reason of "why not" is pretty easy. Jesus wasn't that important at the time. If you weren't a head of state or a Roman politician you would have a hard time getting your name preserved into the present day. It wouldn't be until the 4th century that the Roman Empire converted to Christianity as an example. Recording the life of every "prophet" that popped up in the east wasn't really something most of the empire cared about. The Romans had a very laid back attitude towards religions at this time. As the Christian movement grew, people started to take notice, but it took a while to draw academic interest.

That being said, it is generally agreed upon that there is pretty good evidence for the existence of Jesus given the historical data that we have, even if he wasn't a popular figure for historians during his life.