2 Cor. 5:19-21: “that God was in Christ reconciling the world to Himself, not imputing their trespasses to them, and has committed to us the word of reconciliation. Now then, we are ambassadors for Christ, as though God were pleading through us: we implore you on Christ's behalf, be reconciled to God.  V.21 “For He made Him who knew no sin to be sin for us, that we might become the righteousness of God in Him.”

The righteousness of God is our being given salvation from God, justified through the blood of Christ, his substitutionary death, we are now experiencing sanctification through his Spirit; all righteousness, justification, and holiness, we receive through HIM. We do not have it on our own separate from Him.

The one who was perfectly innocent, gave himself in a voluntary substitution for his enemies; was treated as IF he were guilty of our sins; subjected himself the wrath and punishment for our sin. We who are guilty and who deserve to be punished, were treated, vicariously as innocent; as if we had not sinned. Thus expression of a substitution resulted in salvation for all who believe. Our faith reconciled us to Him.

As Christ did not literally become sin we do not literally become righteous - complete in the process until we are glorified and completed in Him. The fallen race from Adam are "treated as righteous" by God because of Christ's merits, but one day we will be righteous as the new man in Christ; when we are resurrected and immortal without sin in our body. Then we will rule and reign under Him.

Rom 8:10-11 “And if Christ is in you, the body is dead because of sin, but the Spirit is life because of righteousness. But if the Spirit of Him who raised Jesus from the dead dwells in you, He who raised Christ from the dead will also give life to your mortal bodies through His Spirit who dwells in you.”

In the phrase "righteousness of God," refers to the fact that this is his plan of making people righteous, first by justifying them. He was made sin; we are made righteous; he was treated as if he were a sinner, (yet was pure and sinless) we are treated as if we are righteous, (yet were wicked and sinful).

Rom. 3:25-26: “whom God set forth as a propitiation by His blood, through faith, to demonstrate His righteousness, because in His forbearance God had passed over the sins that were previously committed, to demonstrate at the present time His righteousness, that He might be just and the justifier of the one who has faith in Jesus.”

Paul further explains, Rom. 4:5: “But to him who does not work but believes on Him who justifies the ungodly, his faith is accounted for righteousness”

We were made righteous in God’s eyes because of Christ, we have a righteousness imputed to us by God. One day it will be our possession by a completed transformation of mortal to immortal, sinful to sinless. Until that time  “ we through the Spirit eagerly wait for the hope of righteousness by faith.”(Gal. 5:5).:

That is what is spoken by Paul in Rom. 8:17-19: “and if children, then heirs-- heirs of God and joint heirs with Christ, if indeed we suffer with Him, that we may also be glorified together. For I consider that the sufferings of this present time are not worthy to be compared with the glory which shall be revealed in us. For the earnest expectation of the creation eagerly waits for the revealing of the sons of God.”