In Shakespeare's Richard III, Richard (called Gloucester throughout much of the play) is a singular character in a number of ways. From the very first act, Richard is portrayed as a physically disfigured individual who will do anything to achieve his goal - the English throne. Richard lies, cheats, and backstabs, all in the name of that purpose. While no other characters in the play show these particular colors to the extent Richard does, Richard's counterpart, the Earl of Richmond, shares some characteristics with Richard.
As mentioned, Richard, from the opening curtain, actively pursues the English crown. Throughout the couse of the play, he employs various methods to move closer to this goal. He conspired to kill those heirs ahead of him (and ordered their killings), he attempted to marry into the royal line, he served as the protector to the young princes, seeing it as a platform from which to launch himself onto the throne. Richard, after killing her fiance, seeks to gain Anne's hand in marriage, more for its political advantages than for any genuine feeling on his part.
Richard's securing of Anne's hand in marriage, specifically Richard's motivations for doing so, has a great deal in common with Richmond's own marriage to Elizabeth, the older sister to the two young princes. While Richmond may have felt genuine affection for Elizabeth, the political implications of his marriage did not escape his attention. By doing so, with the death of the two young princes, Richmond would be next in line to the throne (through marriage, at least). Both Richard and Richmond take advantage of political marriages as a means to acquire power.
As leaders in the hours leading up to the battle at Bosworth Field, Richard and Richmond share some similarities. The respect that both enjoyed from their supporters (regardless of how that support was earned) says a great deal about the kind of people the characters of Richard and Richmond are. Both realized that the ends justify the means. The major point of difference between them, however, is that Richard does not qualify the point - all ends justify all means. Richmond, however, does seek to qualify the statement somewhat. In the end, both Richard and Richmond play the political game well.
1. They both use killing to achieve their means. However, their attitude towards the killing is entirely opposite. Richard is callous and indifferent about all the murders he orders, even those of his own family members and of the innocent child princes. Richmond deals with the killing that must occur in the Battle of Bosworth field as a necessary evil and shows no enjoyment in it.
2. They both want to marry Elizabeth, daughter of Edward. Again, their attitudes could not be more different. Richard glorifies the incest that he's suggesting when he argues with the previous Queen (mother of Elizabeth) while Richmond wants to marry her to bring peace.
These vague similarities in their character only serve to highlight their differences. The differences in their two characters are an important aspect to the play, both in terms of plot and character revelation.