The fact that Paul considered himself to be an apostle was probably based on the fact that Jesus had commissioned him.   Paul had been on the road to Damascus to punish Christians.  He thought they were trying to destroy the Jewish religion, and Paul was a Jew--a Pharisee.  He had  papers empowering him to punish the Christians.  Or kill them.  He had previously stood by and held the coats--when a mob had stoned Stephen for preaching the gospel. So killing Christians wasn’t outside Paul’s way of thinking. He considered them to be a threat to the nation of Israel. 

But Jesus appeared to him, and from that moment on, Paul was broken, humbled, repentant, and dedicated to Christ. He was given orders from a new high power--Jesus himself--to go to the Gentiles (whom the Jews hated) and give them the words of salvation.  Paul had considered himself elite.  But Jesus changed his entire life, motivation, and purpose.  He was never the same.  He realized he had been terribly prideful and wrong.  God loved the Gentiles as much as he loved the Jews.  God loved all people. 

Peter had asked those waiting on the Holy Spirit, (after the resurrection) to appoint a new apostle--to replace Judas.  Peter took it on himself to do this.  Neither the Holy Spirit, nor Jesus, had empowered Peter to do this, but Peter had an action personality.  The group chose Metthias, and you never hear his name again after that.  Paul most probably felt he was the chosen apostle, appointed by Jesus himself.  When you read a letter written by Paul, most of them will start with words like those in Galatians, “Paul, an apostle, (not by men...) by Jesus Christ and God the Father...”