Richmond only appears in the fifth act of the play, but from that moment his job is to be the sweetest, saintliest thing since, well, the saints. Basically, his goodness is there to put Richard's evil in stark contrast.
Richmond is warm and loving to his supporters, while Richard is demanding and tempestuous with his – they follow him out of fear. Richmond is a careful planner and pious before God, while Richard is brash and overconfident about his own abilities, never appealing to a higher power. Ultimately Richmond's goodness stands in for a more universal goodness – we should see him as the antidote to Richard's evil, for the sake of the play and also for the sake of England's political future. He doesn't need to be a very fully fleshed out character, as he's actually more symbolic of the positive change that's about to sweep over England, a spring to end Richard's winter chill.