To become-a state of self works or self help
To be made-a state of something or someone outside of us, changing us
First let me say that we do not BECOME righteous, we are MADE righteous.
It is done through the perfect attribute of the character of God. That attribute is God's love.
Love found a way whereby God could retain His holiness, His justice, His truth, and yet show mercy to an unrighteous people. His wisdom designed that way, which is the ONLY WAY. His power brought it forth to man. The incarnation of our Lord Jesus Christ was THE WAY to our being made righteous.
As love personified itself in our Lord, God identified with man so that His righteousness could be fully put to man's account. Therefore, man could be called "righteous." In the person of Jesus Christ, God came to man in complete and perfect identification in everything except the sin nature and sin.
God the Father and God the Son remained in a state of righteousness as mankind was being redeemed. So that man might have righteousness or be made righteous, God gave Himself to man in the person of His Son. The Son of God became the Son of Man so that He might redeem the sons of men.
In the person of the Son of God....
Psalms 85:10 ESV Steadfast love and faithfulness meet; righteousness and peace kiss each other.
Therefore, in HIM and through His love, we are MADE righteous.