This is truly an excellent question. One that could actually take pages and chapters to answer (which many have done and I will cite below) but I will do my best to try and help answer it briefly. I have also answered similar questions to this in the past, so I hope you don’t mind if I copy and paste some of my past answers into this thread and just prune it to answer your question.

Right off the back, it can be safely argued that Jesus was pretty unpopular around the time of his death (which we can safely argue was probably around the year 30, plus/minus 5 years) but he would have had to have had at least a somewhat sizeable number of people for his teachings to gain followers. Why do we know that? Well there are a few things that make us think this way.

First, the first person to start writing about Jesus was Paul, who started writing sometime around the year 50. Now it’s important to recognize that Paul never met Jesus while he was alive, so he never writes about many things Jesus did or said. Historian Bart Ehrman actually wrote about this in his book last year: How Jesus Became God: The Exaltation of a Jewish Preacher from Galilee Where in it, Ehrman explains:

The problems with Paul are that he didn’t actually know Jesus personally and that he doesn’t tell us very much about Jesus’s teachings, activities, or experiences. I sometimes give my students an assignment to read through all of Paul’s writings and list everything Paul indicates Jesus said and did. My students are surprised to find that they don’t even need a three-by-five card to list them. — (Page 88).

It is important to note that it is widely believed that other Christians had started writing about Jesus prior to this period, but that their work no longer exists.

So why do I mention this about Paul? Well, it’s because it shows that Jesus’ followers had started to pick up by this time period (20 years after his death). It also shows that by the time Paul started writing, there had to have been a sizable and somewhat diverse group of people who started following Jesus’ message. The writings of Paul are quite well-written (for their time) and the Gospells (which would be written in the following few decades) are even more professionally written for their time (Mark being the least well-written and Luke being the best). This does suggest that in order to get a sizable amount of well-educated people to write down these stories, regardless of their errors, there had to have been a decent size following or their population would have faded away (like many other cults during this time period.)

I will say though, that his followings couldn’t have been too large. I argue this because there isn't a whole lot about Jesus written by non-Jewish/non-Christians (I use this last term loosely when applying it to the first century CE) prior to the second century. As some people may note, the first person to write about Jesus and his followers who was not a Jesus follower was the Jewish historian Josephus, who would only briefly mention Jesus in his twenty plus volume history of the Jewish people called The Antiquities of the Jews. Here Josephus discussed the execution of a man named James, whom Josephus described as “the brother of Jesus who is called the messiah” (Ant. 20.9.1) (I’d also like to note that Josephus never believed in Jesus being the Messiah, he was merely stating what others had said). He does also discuss Jesus again, but I won’t bore you with ancient quotes that would take me far too long put into context.

Long story short: Followers of Jesus would have had to have had a decent number behind them by the time Josephus left the Levant (area near present day Israel) in order for him to have mentioned them at all, but due to the fact that he barely says more than a paragraph about him in all his writings suggest that Christianity (as it would soon be called) was not too popular.

TL;DR: It’s likely that Jesus was not too well-known at the time of his death (the average Jewish peasant was unlike to have heard of him had they not lived un Judea), but his following would have had to have been sizeable enough to spread the stories of his life by oral tradition in order to gain such a noticeable following by the second half of the first century CE. It’s also worth reading John Dominic Crossan’s book: The Historical Jesus: The Life of a Mediterranean Jewish Peasant to gain more information about this very topic.

EDIT to add one more source Another great book that actually deals with the very people we are talking about (early Christians in the immediate aftermath of Jesus' death) was done by an excellent historian named John Dominic Crossan, who wrote a book called: The Birth of Christianity : Discovering What Happened in the Years Immediately After the Execution of Jesus . This is definitely worth a read for anyone who is interested in this topic. I'm ashamed I didn't remember it earlier.