Rivers is first introduced to us in Act I scene 3, and is shown to be part of a trio of characters who are related to and allies of Elizabeth, the wife of King Edward IV and the mother of the two young princes. Rivers in particular has a special link to her as he is her brother. The other characters that he is associated with are Gray and Dorset, who are her two sons from Elizabeth's first marriage. Rivers and Gray are eventually executed by Richard, but Dorset is able to escape and survives.
Rivers is a character who is marked by his goodness and perhaps his naivety and innocence. He is certainly not able to be compared to the Machiavellilan Richard III himself, as his presentation as a godfearing and loyal character shows. Note what he says to Gloucester in Act I scene 3 regarding divine rule:
Against the machinations of other characters, the defence of Rivers is to show his honourable loyalty towards the appointed King at that time, reflecting his belief in divine rule and the way in which Kings were believed to be appointed by God to carry out the job that they were given. This rather feeble explanation of his behaviour shows how unsuited he is to exist in a court atmosphere that is marked by far more sophisticated power struggles.