Can God actually clothe a believer with the righteousness of Christ?

Yes. God imputes His righteousness the moment a person believes the Gospel with the heart. Scripture pictures this as clothing, as when God clothed Adam and Eve with coats of skins and Isaiah spoke of garments of righteousness. Paul confirms the same happens to all who believe.

Les teaches that with the heart man believes unto righteousness, and that this is an imputed thing. As soon as we believe the Gospel, God imputes His righteousness. He connects this to Romans chapter 3, where Paul says the same thing happens unto all that believe. He also illustrates it from Genesis, when God killed animals for Adam and Eve and clothed them, explaining that the coats of skins covered their nakedness but the statement that He clothed them pointed to righteousness. Then in Isaiah chapter 61, Isaiah said, "Thou hath clothed me with the garments of righteousness." So the same truth is repeated: God can actually clothe the believer with the righteousness of Christ.

Romans 10:10

For with the heart man believeth unto righteousness; and with the mouth confession is made unto salvation.

When does God impute His righteousness to a person?

God imputes His righteousness to a person when that person believes the gospel—that Christ died, was buried, shed His blood, and rose again. Just as God saw Abraham believe and counted him righteous, so God credits His own righteousness to believers. It is all of His righteousness, none of our own.