In Richard III, justice is divine and retributive. In other words, every character who has ever sinned or committed a crime gets what's coming to him or her. Since we're dealing with characters who don't hesitate to stab their friends and families in the backs (literally and figuratively), a whole lot of justice is distributed in this play – even for crimes that date back to earlier plays in the tetralogy – Henry IV Parts 1, 2, and 3. (At one point we're told that Margaret is suffering for something awful she did back in Henry VI Part 3.) At times the play even suggests that Richard's entire reign is God's way of punishing all the Lancasters and the Yorks for the Wars of the Roses. By the end of Richard III, Shakespeare has pretty much cleared the decks of all the bad guys, and the play looks forward to a new beginning under Henry VII's reign.
Richmond is the only agent of the play who's depicted as having completely clean hands. Free from perjury, murder, and corruption, he's a golden boy. Accordingly, Richmond is the only one who can act as an agent of justice and finally take Richard down.
The play suggests that Richard is an instrument of divine justice – God uses him to punish the Royal House of Plantagenet for fighting over the crown.