The teaching about righteousness is a Biblical term that occurs in the book of Hebrews.
Of whom [ Melchizedek] we have many things to say, and hard to explain, seeing you are dull of hearing. For when for the time you ought to be teachers, you have need that one teach you again which be the first principles of the oracles of God; and are become such as have need of milk, and not of solid food. For every one that uses milk is unskillful in the word of righteousness: for he is a babe. But solid food belongs to them that are of full age, even those who by reason of use have their senses exercised to discern both good and evil. (Hebrews 5:11-14 KJ2000)
I have written extensively on this subject, particularly in my books When We Awake and Food Sacrificed to Idols. Given the great apostasy of the church as clearly seen in other links on this site, and as personally experienced on The Shack Forum the last three days, we would do well to review it.
First, Hebrews 5:14 counsels mature Christians to become able to discern between good and evil. Anyone who cannot do this remains a babe in Christ, unskillful in the word of righteousness. This means that such a one cannot understand the Word of God, the Bible. Oh, he will think he can, but his words and lawless attitude betray him as a mere babe.
The teaching about righteousness has nothing to do with obeying a particular set of principles, commonly called The Law, through the power of the flesh. All of history shows that mankind cannot perfectly obey God’s Law. This teaching also has nothing to do with church-made laws or man’s laws. Instead, the teaching about righteousness is all about being led by the Holy Spirit. Through his leading and by the washing of the water of the Word, we learn and are enabled to apply God’s principles to all of life. Thus moral principles like forsaking fornication certainly apply to us in the natural as written in Exodus 20. On the other hand, by following the Spirit we understand that we do not need to continue sacrificing bulls, sheep, goats, or birds according to the book of Leviticus. This is because Jesus himself fulfilled all of those sacrifices and we apply his sacrifice to our life by faith in him.
Jesus himself teaches us to be perfect, not perfect by some human standard, but perfect as his Father in heaven is perfect. God calls this perfection “righteousness.” Another word for this is “holiness.” Our God IS righteous and holy and fully expects us to be too. Jesus did not make some flippant, unattainable comment when he said, “be perfect as my Father in heaven is perfect.” He meant it.
So how do we “be perfect.” First, become acquainted with God’s ways. Get to know his definitions of right and wrong. Learn how to apply these principles to all of life’s many situations. Second, begin to desire to obey particular standards and rules of conduct. This has to do with taking every thought captive to Christ so that you will obey his will instead of your own which has been steeped in sin. Of course this obedience does not “earn” your salvation. Only an imbecile would suggest that it does once he understands that Jesus died for us and forgives us our sins. Third, repent of anything you do that you know is contrary to God’s will. Anything that is not of faith is sin, and don’t fool yourself into believing that your clearly sinful acts are done in faith. I’m afraid that won’t work with God.
John said it clearly, “No one born of God makes a practice of sinning, for God’s seed abides in him, and he cannot keep on sinning because he has been born of God.  By this it is evident who are the children of God, and who are the children of the devil: whoever does not practice righteousness is not of God, nor is the one who does not love his brother.” (1 John 3:9-10 ESV) So, you may disagree and say that God does not call us to be righteous and holy, but you will be wrong. You would be in agreement with Satan instead of God.
The Old Testament proves we cannot perfectly keep God’s Law. God knows you are not inherently perfect. That’s why he gave us an escape valve from sin. It’s called repentance. That’s why its the first elementary doctrine of the Christian faith mentioned in Hebrews 6. So long as we faithfully seek to obey Christ and repent when we fail, we maintain our perfect standing before God. This is all God asks of us. [Next]