I think that Jesus' was reassuring the Jewish people that he had come to fulfill the prophecies of the Old Testament. "Righteousness" can be interpreted different ways in Christianity -one of the complexities of the Bible is that in the Old Testament God used people with obvious sins to serve him. The one thing all of these people had in common was their faith and love of God, not blamelessness. No one can be blameless, because everyone is human. That is why we are all equal in the eyes of God and forgiveness is a requirement.
Most Christians agree that the ten commandments are the law that should not be broken, which Jesus sums up in the two commandments to love God above all others and to love others as one's self, as wildcat already mentioned.
As for all men being born "evil" we all are born with certain instincts that make us selfish, even cruel. Ever seen a two year old share a favorite toy without being taught to do so?