Paul writes to Timothy to flee and pursue, and gives a list of things to pursue. That list begins with righteousness. Paul gives us an interesting dilemma here. There is no one who is righteous, no not one. It is only in Jesus that anyone is righteous, since none of us have a righteousness of our own. So what is Paul telling Timothy? Why would he tell him to pursue something that is given by God, not earned in some life pursuit? I think for very good reason.
The word righteousness in the Greek is the same word as justice. We translate it both ways, but in English we use those words differently. In reality, they are two sides of the same coin. We can look at it and say that because God is just, He must require that we are righteous. His justice requires our righteousness. He requires us to be like Him. But knowing we cannot ever be like Him, He provided Jesus to pay the penalty for our sin, thus making anyone who is in Christ a new creation, and righteous.
But Timothy was already in Christ, so why would Paul tell him to pursue righteousness? We are called to continue to pursue holiness, working out our salvation, as long as we are on this earth. We are to continually be transformed to more fully reflect the light of Christ to the world. We are to press on toward that goal, pursuing it as we would pursue a prize at the end of a race. The danger is that if we are not pursuing righteousness we will drift toward unrighteousness. That is our nature. That is our flesh. That is the danger of lukewarm Christianity. We must pursue righteousness, constantly striving for obedience, not that we earn salvation through it, but that we would be more faithful disciples. This is what it means to delight yourself in the Lord and nothing else. Constant pursuit of Him!