circumcised on the eighth day, of the people of Israel, of the tribe of Benjamin, a Hebrew of Hebrews; as to the law, a Pharisee; (ESV)
If you were to ask people on the street why they believe God should allow them into heaven when they die, you will probably get a majority of people listing for you all kinds of "good deeds" they have done that somehow merit eternal life. They may cite money they have given to feed the poor, time spent with underprivileged kids, or even church attendance.
Paul is dealing with people who believe that because of their Jewish heritage they have an in with God and are deserving of eternal life. These Jews have a big problem with Gentile Christians who claim to have eternal life simply by believing in Jesus Christ while not embracing all of the rites and rituals of the Jewish faith. Paul now says to the Jews, "You think you are righteous based on your Jewish background? If it was about that I would be in way before you all!"
Paul is a circumcised Jew. He is of the small, favored tribe of Benjamin. He describes himself as a Hebrew of Hebrews. And probably the most significant aspect of his credibility was his being a Pharisees. The Pharisees were the teachers of Jewish Law. Here is how Jesus himself described the Pharisees:
 They tie up heavy burdens, hard to bear, and lay them on people's shoulders, but they themselves are not willing to move them with their finger.  They do all their deeds to be seen by others. For they make their phylacteries broad and their fringes long,  and they love the place of honor at feasts and the best seats in the synagogues (ESV)
Paul was a Pharisee. He knew about all of the burdens of legalistic righteousness. He had performed all the rites and rituals. And he knew it wasn't enough. Same is for you. Without the work of Christ on your behalf you will never see heaven.