I understand his post-conversion contributions to the church according to the NT. But what I'm wondering specifically is how he went from persecuting Christians to being one of their greatest champions. If he was indeed persecuting Christians, he would need to build a huge amount of trust after the conversion. How did the early Christians trust him post-conversion, especially if he really was persecuting them? Was the persecution not as bad as it sounds? Was he a really good "politician?" According to Acts, he was a strong preacher and had other apostles and disciples such as Ananias vouch for him. Was this all he needed to dissipate mistrust?
Do we have any outside of the Bible sources that can get us close to what actually happened regarding Paul's acceptance in the early church (in the context of his persecution and conversion and assuming the historical figure is close to who we read about in Acts, etc.)?
Short answer: no. We really have almost nothing to go on.
One thing to note, which isn't clear from Acts, is that it's generally accepted there is a big time gap between Saul's conversion and the start of his major ministry: the former is put at some point in the 30s, and the start of the first missionary journey usually in or about 48. It's unclear what Saul was up to for most of this period (there are some tentative suggestions) but presumably Christians had time to accept him as genuine.
To add to this: there are clearly some tensions continuing between Paul and the Jewish Christians (see Galatians 2), and remember that a large part of the early Gentile church was founded directly or indiirectly by Paul.
But in terms of sources outside the Bible, we have nothing at all contemporary that I'm aware of, and I don't think any stories survive in early Christian writings.
As for outside sources, some of the early Church fathers (Polycarp, et al) may have some insights, but I don't know of any offhand. However, in Acts Paul's influence comes from his meeting with the disciples in Jerusalem post conversion via Barnabas (Acts 9).
Well, we must remember, according to Acts, that the church leaders did not at first trust him. Paul mentions to the Galatians he initially traveled to Arabia, tradition dictates to fast and pray then went home to Tarsus for 3 years and only emerged when he wanted to spread the gospel through mission, it seems it took this long to be actually endorsed by the church in Jerusalem. As for Ananias, it seems he genuinely believed him to be a changed man, no external sources that I can find mention him, but he must have been terrified on first meeting Paul.
As things go he was probably already well respected among Jews as a notable lawyer and possibly a member of the Sanhedrin. As a famous antagonist to Christians his conversion would have been high profile, like say the conversion famous atheist scientist would be. whats more he was an educated Pharisee, unlike the disciples so he provided a detailed analysis of Jesus as the Messiah from a Jewish perspective, this would be how he got such an audience.
Once endorsed with the gospel, he went out church planting. Thus he would've got most of his popularity in Greece where Christians did not experience the Pauline Persecution. As for the severity of the persecution, at this time the Jewish authorities were committed to destroying the new sect, so it was probably severe.
As for extra biblical sources, Polycarp speaks very favourably of Paul, and notes John did too, but obviously this is all spoken with hindsight. Josephus records many of the events of Paul's life, but doesn't really get into his initial relationship with the church. Some fragments survive of Clement of Rome's letters, but they mostly pertain to defending Paul's apostleship in their references to him post conversion. really if you want to know about Paul, the best place to look is the NT, it may be religious but it is still a historical text and shows the story from Luke's, Paul's and other early christian perspectives.