The conversion rate to Christianity was not rapid in the first few centuries after Jesus' death. I think it best to think of it as a steady rate, but when the base number for the rate was increased so was the number of converts. (Check of the book, Triumph of Christianity - Stark (I think); or Harnack's Mission and Expansion..Bart Ehrman is also coming out with a new book on the topic this year or next.)

Anyways, Jesus was likely relatively famous within Judaea as a preacher, with a reputation for miracles, and perhaps as a sort of political reformer (as Judaea's politics and religion were intertwined in Judaism). He almost certainly thought the world was nearing its end, but so did other Jews in his lifetime. He was by no means empire-wide famous, and in reality, he likely was more famous in Galilee than anywhere else. But figures like this did gain regional followings, Josephus mentions several similar figures in his Jewish War. Anyways, he does seem to have gathered quite a following, as did John the Baptist.

After his death, his popularity shot way down. Remember, if any Jews did think he was the messiah, they didn't when he died in one of the worst ways in the ancient world. The messiah was not supposed to die. Even in the Bible, his crowd dispersed upon his arrest and trial. Afterwards, the Gospels mention at most 20 or so followers.

The big question is, and the one Stark, Harnack, Ehrman, and others are trying to answer is how did this minuscule group of followers grow to become dominant by the late 5th century.

Edit: more proof for his lack of popularity outside Judaea, after his death, is the utter lack of Jewish and Pagan sources on him after his death. Within he first 100 years, he's only mentioned a handful of times, and the pagan sources do so indirectly.

Edit 2: I forgot your last question. We don't know how much leaders we're actually aware of Jesus during his preaching throughout the region. However, this changes when he goes to Jerusalem for the festival. He apparently caused a scene in or around temple courtyard, and was betrayed by Judas Iscariot to some capacity. More importantly, upon entering Jerusalem, he quickly gathered a following (likely because of his preaching and miracles). Now the primary job of the Roman officials and Jewish leaders during festivals was to maintain peace. Jerusalem had a bad record of maintaining peace during festivals, so it was a pretty high alert time. If a popular preacher did in fact preach against the temple and Jewish officials, this would have brought much unwanted attention.

Azlan's Zealot is a basic introduction to Jesus's possible political views. Hyam Maccoby had a great one (Revolution in Judea) of a similar vein.

***Above all, I recommend E.P. Sanders' The Historical Figure of Jesus. It is a must read, and very short.