By what is said in the Bible, righteousness is wisdom of the just, right understanding that leads person to right actions.
He will go before him in the spirit and power of Elijah, 'to turn the hearts of the fathers to the children,' and the disobedient to the wisdom of the just (=righteous); to make ready a people prepared for the Lord.
Luke 1:17
One can get that right understanding for example from the words of Jesus. The words can cause change in person so that he becomes righteous, gets the right understanding. And when person has the right understanding, he freely wants to do what is good and right.
I think these scriptures speak about the change in person to become righteous:
Jesus answered him, “Most certainly, I tell you, unless one is born anew, he can’t see God’s Kingdom.” Nicodemus said to him, “How can a man be born when he is old? Can he enter a second time into his mother’s womb, and be born?” Jesus answered, “Most certainly I tell you, unless one is born of water and spirit, he can’t enter into God’s Kingdom. That which is born of the flesh is flesh. That which is born of the Spirit is spirit. Don’t marvel that I said to you, ‘You must be born anew.’
John 3:3-7
But as many as received him, to them he gave the right to become God’s children, to those who believe in his name: who were born not of blood, nor of the will of the flesh, nor of the will of man, but of God.
John 1:12-13
It is the spirit who gives life. The flesh profits nothing. The words that I speak to you are spirit, and are life.
John 6:63
He who does righteousness is righteous, even as he is righteous. He who sins is of the devil, for the devil has been sinning from the beginning. To this end the Son of God was revealed: that he might destroy the works of the devil. Whoever is born of God doesn’t commit sin, because his seed remains in him, and he can’t sin, because he is born of God. In this the children of God are revealed, and the children of the devil. Whoever doesn’t do righteousness is not of God, neither is he who doesn’t love his brother.
1 John 3:7-10
“For this is the covenant that I will make with the house of Isra-el. After those days,” says the Lord; “I will put my laws into their mind, I will also write them on their heart. I will be their God, and they will be my people. They will not teach every man his fellow citizen, and every man his brother, saying, ‘Know the Lord,’ for all will know me, from their least to their greatest. For I will be merciful to their unrighteousness. I will remember their sins and lawless deeds no more.
Heb. 8:10-12 (Jer. 31:31-34)