First, in scene i, Buckingham is executed. Even though he and Richard were joined by a common purpose at the beginning of the play, Buckingham refused to kill Edward IV's children....
In Shakespeare's Richard III, there are several important things that take place in Act V of the play.
First, in scene i, Buckingham is executed. Even though he and Richard were joined by a common purpose at the beginning of the play, Buckingham refused to kill Edward IV's children. Abandoning Richard, Buckingham joined Richmond, but has been captured by Richard. Prior to his death, Buckingham recalls all those who have died, imagining that if they can see the end he has come to, they must be enjoying the moment:
If that your moody discontented souls
Do through the clouds behold this present hour,
Even for revenge mock my destruction! (7-9)
In scene ii, Richmond announces that their progress toward Richard has not met with resistance. He has had words of encouragement from Stanley, Richmond's step-father...
And here receive we from our father Stanley
Lines of fair comfort and encouragement. (5-6)
Richard has tried to control Stanley's behavior by taking Stanley's son, George, hostage; Richard has threatened to kill George if Stanley take up Richmond's cause, so Stanley must be very careful in supporting Richmond.
In scene iii, Stanley commits himself (secretly visiting Richmond in his tent) to do whatever he can to assure that Richard falls to Richmond in the upcoming battle.
I, as I may,--that which I would I cannot,--
With best advantage will deceive the time,
And aid thee in this doubtful stroke of arms:
But on thy side I may not be too forward,
Lest, being seen, thy brother, tender George,
Be executed in his father's sight. (95-100)
Stanley wishes he could do more for Richmond, but reminds him that George is at Richard's mercy, so Stanley cannot support him openly for fear of George's life.
Scene iv finds Richard fighting like like a madman, on foot because his horse has been killed. It is in this scene that Richard cries out the famous line:
A horse, a horse! My kingdom for a horse!-- (7)
Finally, at the end of Act V, in scene v, Richard and Richmond face each other on the battlefield. In the battle action, Richard is killed, and Richmond praises his men and declares that the day is theirs: they have won the battle.
God and your arms be prais'd, victorious friends;
The day is ours, the bloody dog is dead. (1-2)
We also learn that Stanley's son, George, has been unharmed.