The way of righteousness is by divine initiative. He says that Christ is the fulfillment of the law so that there may be righteousness for everyone who believes. God gave Israel the Law, not so they could keep it in every aspect, but so that the Law could reveal to them their inability to keep the righteous requirements of the Law. [Oh, please read that again: God gave Israel the Law, not so they could obey it to the most minute particle, but so that the Law could reveal to them their inability to keep the righteous requirements of the Law]. Israel knew they could not keep it. However, they did not see the Gospel when it came in the person of Jesus Christ.
Paul did not simply say, “Christ is the end of the Law.” Many Christians today like to say that we are “not under Law but under Grace.” When we do that, we are taking this Scripture out of context to mean that we do not have to take the commandments of the Old Testament seriously. It is as if the Old Testament does not count any more. Some Christians do not even read the Old Testament because they think that it is Law and the New Testament is Grace. The Law has never been done away with, in terms of its application to Christianity. All of the ceremonial law and all of the extra-biblical interpretation that Jews placed on the law are gone. That was never to apply to us anyway. However, laws like, “Thou shalt not steal,” “Thou shalt not murder,” “Thou shalt not covet,” still apply today. The Law is an expression of the heart of God. The Law was never unrighteous. Mankind is unrighteous in that we can not keep the Law. The Law is righteous, holy, and just. Paul has already made that point in this letter.
There is nothing wrong with the Law. Today, we have the power to keep the Law in the Person of Jesus Christ. As Jesus Christ is allowed to live his life through us, he will cause us to keep the righteous requirements of the Law. He will cause us to love God with all of our minds, all of our souls, all of our strength. He will cause us to love one another as well. These are the two Great Commandments. Jesus said that upon these two commandments hang all the Law and the prophets. If you are doing these two, you are going to be keeping the Law. You are not going to be coveting, stealing, lying, or murdering if you are really sold out to God. However, it will be Jesus Christ keeping the Law in you and through you.
What Paul is saying is that we do not attain heaven through keeping the Law. We attain heaven through coming to Christ. The way of righteousness is not by human endeavor, but it is by our response to his divine initiative.
If interested, you can download the entire study of The Letter to Christians at Rome