From the moment we meet Richard, we learn that he's dissatisfied. He announces that he's "determined to prove a villain." We then get the outline of how this will happen: as Richard chats with a prison-bound Clarence, it becomes clear that he has no qualms about deceiving and conspiring against people. He hints at his full plan once Clarence is safely in jail. He'll have Clarence killed, wait for Edward to die, then marry Anne. While Richard details his plot to us, he's still careful not to celebrate it, as a lot of things need to fall into place before he has a chance at the crown.
Everything is going swimmingly for Richard. With both his brothers dead, his path to the throne is only blocked by two little boys, whom he has decided are sniveling and inconsequential little kids. Richard has also successfully removed Queen Elizabeth's family from the picture. Rivers, Gray, and Vaughn were politically irksome to Richard, and they also provided some protection to the young princes. With the protectors out of the way (and Queen Elizabeth's sanctuary for her young son disabled), Richard has unfettered access to both boys, meaning he can get them out of the way easily.
Richard has to figure out who's with him and who's against him. Hastings is not an easy sell, and Richard has him killed easily enough, but he has to cover his tracks to the mayor and the people. Not surprisingly, the citizens are not at all enthusiastic about Buckingham's suggestion that Richard should be king. This frustrates Richard, and though he gets the crown anyway, he's in a pretty foul mood about the whole thing.
Even after he's been crowned, Richard is worried about maintaining his power; he has to have the little princes in the tower murdered. When Buckingham hesitates on this point, Richard dismisses him without a second thought. This might be Richard just being rash, but it also might indicate that Richard knows that his position is precarious: he has no time to appease people who don't support him wholeheartedly, because he's beginning to gather a crowd that opposes him wholeheartedly.
Richard is already receiving word of desertions to Richmond's side when Buckingham readies to leave him. Weirdly, Richard's calm (if glum) reaction to all of this is a reference to a prophecy he once heard that Richmond would be the end of him. He's not yet in a panicky stage, and he's still making plans to seal up his power by marrying young Elizabeth. He also gets the comforting news that his nephews have been smothered to death.
The situation begins to worsen when Richard learns that Richmond is being joined by more powerful forces than Richard had anticipated. After Richard's mother curses him to die a bloody death, things just keep going downhill. Richard really seems to break when he learns for certain that Richmond's navy, fat with Richard-deserters, is headed for Richard's shores.
For the first time, Richard appears disoriented and confused. He's not used to losing control, and his inability to hold himself together doesn't bode well. The best evidence that this is Richard's nightmare stage is that he, in fact, has a literal nightmare. He's confronted by the ghosts of everyone he's had a hand in murdering, and they all condemn him to die in the next day's battle. (Just so we're clear on Richard's inevitable end, the ghosts also visit Richmond and encourage him to defeat Richard.) Richard awakens and is visibly shaken. He can't seem to get a grip on who he is or what he's about.
Richard keeps receiving news of his supporters deserting him, and he seems to content himself with the declaration that he and his men will face whatever is coming to them head on. When Richard says "March on, join bravely, let us to it pell-mell; If not to heaven, then hand in hand to hell," it's a pretty good indicator that he's ready to meet his end. He's determined not to be cowed, and he'll go down fighting. Richard has no regrets, and his last-ditch effort is to try to kill Richmond. His final lines on the stage illustrate that he's ready to face the consequences of all the chances he's taken, even if he knows the odds are against him.