Righteousness is the quality of being morally right or justifiable, living in accordance with God's standards and His will. Pursuing righteousness does not mean becoming self-righteous or trying to be good enough. No one is righteous in themselves and no one can be justified through obedience to the Law. Rather, true righteousness is responding properly to having right standing before God. Only He can make that happen for us. The fruit of righteousness is a work of the Spirit of God in our lives, yet it is also something we are commanded to pursue through following God's ways.
God initiates a relationship with us through Jesus who paid for our sins. Because He did, those who put their faith in Him are considered blameless before God when it comes to eternal judgment. We are also made new and given the Holy Spirit. The Holy Spirit works in our lives to sanctify us, which is to make us righteous in practice. As we grow in our relationship with God, we become more like Him. It is in living out the reality of our salvation that we pursue righteousness. This happens through sanctification and ultimately glorification.
The New Testament ties righteousness closely with faith. For example, Romans 4:13 says, "For the promise to Abraham and his offspring that he would be heir of the world did not come through the law but through the righteousness of faith." We must take care not to get things out of order. We are made righteous by Jesus and through putting our faith in Him, not through our own works (Ephesians 2:8–10). Our response to being made righteous is to act accordingly. We don't behave righteously to earn salvation or standing before God. Rather, we pursue righteousness because of our new identity in Christ (Ephesians 4; Colossians 3:1–17; 1 Peter 1).