With regard to the first task, it may well be the case that the specific phrase "the righteousness of Christ" is not found in the scriptures. It is also the case, though, that the scriptures speak of the justification of Christ. First Timothy 3:16 says clearly that Christ was "justified in the Spirit." This action is referring to Christ's resurrection where God overturned the verdict of the human courts and declared Jesus to be justified before the heavenly court. One who is justified is righteous. So, even though the precise phrase "the righteousness of Christ" is not in the New Testament, the scriptures certainly speak of the resurrection as constituting the verdict that Jesus is indeed uniquely righteous.
With regard to the second task we turn to those passages of scripture which speak of the believer being united with Christ in his death and resurrection. One such passage is Romans 6:5, "For if we have been united with him in a death like his, we will certainly be united with him in a resurrection like his." The language of "union" here is the language of covenant. With his death and resurrection, Christ inaugurated a new covenant, of which the benefits become ours when we are united to him through faith. Thus, if we are guaranteed final resurrection by virtue of our union with Christ, because what is true of him is true of those who are united with him, then we are guaranteed the final declaration of justification by virtue of our union with Christ. His resurrection guarantees our resurrection and our final justification. Present justification by faith is an anticipation of that future and final justification. This should not be thought of as two verdicts, by the way, but the one verdict of the future realized in the present through faith. The point is that the unique righteousness that belongs only to Christ by virtue of his resurrection is shared with those who have faith-union with him such that they too can be said to be righteous or justified. And what is the basis of this justification? It is nothing other than the declaration of righteousness granted to Christ at his resurrection because of his obedience unto death which is then transferred to believers through union with him. And what is the word used to describe this covenantal transfer of righteousness by virtue of faith-union with Christ? It is nothing other than imputation.
To summarize the argument, we can say that we are justified through the imputation of the righteousness of Christ because Christ's resurrection constitutes the verdict of God that Christ is indeed justified, or righteous. Through faith-union with Christ, his unique righteousness is granted to believers precisely because we share all that is his in union with him. Our resurrection from the dead will be the ultimate realization of the righteousness of Christ imputed to us. This is why Paul can write that Christ was raised for our justification (Romans 4:25). Christ was justified in his resurrection. Insomuch as we are joined to him, the verdict of righteous that is his in the resurrection is imputed to us through faith.
All this is phrased with excellence in the final verse of Charles Wesley's hymn "And Can it Be that I Should Gain." The emphases are mine, of course.
No condemnation now I dread;Jesus, and all in him, is mine;Alive in him, my living head,And clothed in righteousness divine,Bold I approach th'eternal throne,And claim the crown, through Christ my own.Bold I approach th'eternal throne,And claim the crown, through Christ my own.