For I tell you, unless your righteousness exceeds that of the scribes and Pharisees, you will never enter the kingdom of heaven.
Let's begin by forgetting what we know of the scribes and Pharisees for a moment, and think about what the hearers of this might have thought.
Scribes and Pharisees were the religious cream of the crop. These were people who had devoted thier lives to the reading, understanding, and practice of the Torah. They not only knew the rules, they interpreted and enforced them too. Think of it as congress, the Supreme Court, and the President all rolled into one. Their religion was constantly on display, from the clothes they wore and the accessories on those clothes to the places they went to the people they associated with. Scribes and Pharisees were righteous dudes, all the way. They were able to devote all of their time to their pursuit of God. And Jesus just said, unless you can do it better, you won't even get into the kingdom.
So I see two options here. The first is that I have to be better, to spend more time than they spend, and to make my efforts greater than theirs. That does not sound possible, so option two: consider the fact that perhaps the scribes and the Pharisees were not as righteous as they seemed. But even that option does not seem to make a lot of sense. How could these men not be righteous?
Suddenly, option three comes into play. Maybe Jesus was talking about a different righteousness. Or a righteousness that comes from somewhere else. Or both. Philippians 3:9 sheds some light on this idea. "and being found in him, not having a righteousness of my own that comes from the law, but that which comes through faith in Christ, the righteousness from God that depends on faith." So it is not about the show. It is not about what I can do. It is all about Jesus, just trusting in Him to do what I cannot do no matter how much effort I put in. This is not an excuse for laziness, but it shows that whatever is lacking in me, Christ can provide as I trust in him to do so. So I don't so much worry about pursing my righteousness as I do pursuing Christ himself.