Why did Paul constantly have to defend his apostleship in his letters?
Paul constantly defended his apostleship because wherever he ministered he faced a recurring slur that he was an imposter, a renegade Jew, and not a true representative of Judaism. His early practice of going first to the synagogue, then to Gentiles, provoked rejection and suspicion, so he repeatedly had to affirm that he was God's man for the hour.
Les explains that Paul repeatedly defended his apostleship because of the persistent opposition he met. In his early ministry Paul always went first to the Jew, to the synagogue, where he could expect at least a halfway civil hearing; when the Jews rejected the message, he turned to the Gentile area and marketplace. Wherever he went, however, there was a constant slur that he was an imposter, not a true representative of Judaism, but a renegade Jew. Therefore he continually had to come back and defend the fact that he was God's man for the hour. Les also points out that this defense arises as Paul addresses the Corinthian church's divisions. The Corinthians were carnal, not a spiritual church like Ephesus or Philippi, and beset with all kinds of problems. Some said they followed Paul, some Apollos, some Peter, and some even Christ. As Paul addressed that division, he again had to defend his apostleship. Though they were carnal, they were still co-laborers with the Apostle, just as believers today are co-laborers together to bring to fruition the eternal will of God.