More righteous than the Pharisees

Matthew
Matthew 5:20 can be confusing unless one digs a little deeper.  The text says: "For I tell you that unless your righteousness surpasses that of the Pharisees and the teachers of the law, you will certainly not enter the kingdom of heaven."  How can our righteousness surpass the Pharisees?  They're experts in following the law. 
The catch is, Christ came to fulfill the law (as mentioned just previously to verse 20).  We can't have a righteousness of our own just by following the law, because we can't keep the whole law.  Fortunately, through Christ, who has fulfilled the whole law, we can obtain righteousness greater than that of the Pharisees.  Faith alone will get us there, not any works that we could do ourselves.  Thus, through Christ, we can enter the kingdom of heaven!