"When did Christians begin calling themselves Christian?" Short answer: it's hard to say, but probably in the early second century. There are two relevant biblical passages. 1 Peter 4:16 (NRSV) says, "Yet if any of you suffers as a Christian, do not consider it a disgrace, but glorify God because you bear his name." The preceding verse is crucial--"Let none of your suffer as a murderer, thief, criminal, or mischief maker." The parallelism suggests that "Christian" is syntactically equated to the list in v. 15, which further casts "Christian" as a negative. The second and third clauses bear this out--IF you're suffering (in this context, probably social ostracism, not violent persecution) as a Christian, it's cool because that's God's name. If the name Christian is considered negative to the point that someone has to convince you not to feel bad about it, the implication is that you're not calling yourself that.

The second verse is Acts 11:26, where we learn that antioch was the place "where the disciples were first called Christians." Note the passive. Again, they may not be calling themselves that. In fact, Acts doesn't call them Christians. It calls them things like disciples, apostles, and followers, and refers to what we would call "Christianity" as "the Way." The Antioch line is kind of a throwaway. Likewise, Paul in his letters refers to being "in Christ." He never says Christian or Christianity.

Some dates: Paul's letters are all between roughly 45 and 65 CE. 1 Peter is from the 60s or the 90s, depending on who you ask, and Acts is from the 90s or roughly 110, depending on who you ask. For reference, Jesus world have been crucified in the 30s. Regardless of opinions on earlier or later dates, the word Christian doesn't appear in the new testament until at least the 60s, and at least as late as the 90s is still and outsider and possible derogatory term. The first insider use of "Christianity" is by either Ignatius or polycarp (sorry, I can't find the exact citation), and is widely considered a novel coining in that letter. That would have been in the first half of the second century.

Now, just because they weren't calling themselves Christians didn't necessarily mean that they thought of themselves as Jews. As you can see from above, they had a number of names for themselves, not all of which are represented in the NT. It's a huge ball of bees that I'm not going to do any justice to in this space, but I encourage you to look into literature on "the parting of the ways." Some of them definitely did think of themselves as Jewish. The inner circle (Peter, James, John, Andrew, Philip, etc.) did. Paul certainly did (e.g. philippians 3:5). But after the first couple of decades, the majority of people joining the movement seem to have been non-Jews. There's a lot of disagreement, in the original sources and in modern scholarship, over how necessary it was for them to actually convert to Judaism. Paul says not at all, others say absolutely. Eventually Paul's perspective won, so that's where the majority of the surviving sources land, but that's not necessarily reliable for actual demographics. A GREAT book for this is Paul: The Pagans' Apostle by Paula Fredriksen. She's focusing on Paul's reasoning, but it tells you a lot about the people he was preaching to.

The disagreements about whether and how much Christians were Jewish go deep into late antiquity. We still have Christian bishops getting angry that Christians are also going to synagogues at least in the fourth century (but that's long after my expertise runs out so I'm not going to butcher it).