This phrase "Righteousness of God." Is intended to indicate all three perspectives: The righteousness which characterizes God, the righteousness imparted by God to believers, and the justification performed by God. So the effort will be to show exegesis for each perspective.
The righteousness stabilized two misunderstood foundations
Foundation 1: Righteousness by faith instead of eating the fruits of the knowledge of good and evil
For when we were in the flesh, the sinful passions which were aroused by the law were at work in our members to bear fruit to death. (Romans 7:5 NKJV)
For the Law is built off of wisdom, and wisdom from knowing good and evil.
So when the woman saw that the tree was good for food, that it was pleasant to the eyes, and a tree desirable to make one wise, she took of its fruit and ate. She also gave to her husband with her, and he ate. (Genesis 3:6 NKJV)
Therefore faith seems foolish to those that are wise.
But God has chosen the foolish things of the world to put to shame the wise, and God has chosen the weak things of the world to put to shame the things which are mighty. (1 Corinthians 1:27 NKJV)
Foundation 2: Righteousness through the communication process
You shall have no other gods before Me. (Exodus 20:3 NKJV)
Although this translation is accurate, it fails to portray the full meaning behind this statement. For God is the Word.
In the beginning was the Word, and the Word was with God, and the Word was God. (John 1:1 NKJV)
What does it mean to have no other Gods before the Word?
But Jesus said to him, “No one, having put his hand to the plow, and looking back, is fit for the kingdom of God.” (Luke 9:62 NKJV)
There are many teachings that Jesus did specifically to help in the understanding of the first set of Ten Commandments, so to avoid getting to far of track.
Perspective 1: The Righteousness which Characterizes God
Therefore, just as through one man sin entered the world, and death through sin, and thus death spread to all men, because all sinned— 13 (For until the law sin was in the world, but sin is not imputed when there is no law. 14 Nevertheless death reigned from Adam to Moses, even over those who had not sinned according to the likeness of the transgression of Adam, who is a type of Him who was to come. 15 But the free gift is not like the offense. For if by the one man’s offense many died, much more the grace of God and the gift by the grace of the one Man, Jesus Christ, abounded to many. (Romans 5:12-15 NKJV)
Before the event of the Fruit of the knowledge of Good and Evil Their where two that had no sin. God and also Adam. The Character of Adam before the fruits was in God's image. So before the fruits there were two that walked the perfect way. Once Adam ate of the fruits, Only God walked the perfect way. Therefore the description of God's Righteousness Characterizes God. Because God is the example to follow.
Perspective 2: The righteousness imparted by God to believers
Because the law brings about wrath; for where there is no law there is no transgression. (Romans 4:15 NKJV)
If one does not consider a situation to be in dysfunction, then function is no longer necessary to develop. Likewise, when function is in process, another function may serve that purpose. Or the function itself could be considered dysfunctional. From a certain point of view. Since the Sin is to eat the fruits of deciphering function and dysfunction. Faith in God is the answer to this dilemma. For when a dysfunctional situation is handed to God. God then delivers. So since the work is done By God it is imparted by God to believers.
Perspective 3: The justification performed by God
For the wages of sin is death, but the gift of God is eternal life in Christ Jesus our Lord. (Romans 6:23 NKJV)
A penalty is only given to those that transgress a law. For those that no longer transgress the law. Mercy is a gift. This mercy is an action by God so the justification performed by God