Richard III is about the struggle to get and hold on to political power, a topic Shakespeare returns to repeatedly in his history plays and tragedies. On the one hand, the play portrays Richard as a "Machiavel," an unscrupulous ruler who'll do just about anything to gain the crown and remain in power. Richard's antithesis is Richmond/King Henry VII, a monarch divinely appointed by God whose reign marks a fresh start for war-torn England. Because Henry VII's reign marks the beginning of the Tudor dynasty, Shakespeare seems to be celebrating his own Tudor monarch, Queen Elizabeth I. At the same time, power can also operate on a more personal level: Richard's path to the throne involves several small-scale power struggles between him and his adversaries.
Richard is not driven to be King of England because of the power it promises. He is actually more delighted with the wickedness he'd have to perform to get there.
Personal politics trump power politics for Richard.
The play is a criticism of Machiavellian power politics. Shakespeare makes Richard an interesting character but ultimately punishes him.