What does the play Hamlet have to say about political power and the use/abuse of such power?

Political power is shown throughout Hamlet as something that can easily be abused. Since Claudius took over the throne after murdering his brother, Denmark has been plunged into a morally rotten state, so much so that the very air that Hamlet breathes is a "foul and pestilent congregation of vapors."
The whole nation has become decadent, as demonstrated by the nightly drunken parties that take place at Elsinore, much to Hamlet's disgust. Claudius is not exercising political power responsibly, and this lack of dedication to his role filters down through his courtiers and sets the general tone for the population at large.
Hamlet knows that things can be so much better, but he's not the one to provide Denmark with the leadership it so desperately needs. Aside from his constant vacillation, Hamlet lacks the moral fiber necessary to restore the nation to what it once was when his father was king. Despite his constant ravings against Claudius, he has been corrupted by his wicked uncle's abuse of political power as much as anyone.
His feigned madness, which eventually accounts for the deaths of four people—Polonius, Ophelia, Rosencrantz, and Guildenstern—is nothing more than a game of political chicanery—the kind that Claudius himself would probably play were he in Hamlet's position. Hamlet may see himself as occupying a higher moral plane than Claudius, but in actual fact, he's no less of a scheming politician. It's just that Claudius' tactics are different.
It's instructive that it's only with the arrival of a foreign ruler, Fortinbras, that some semblance of order is finally restored to this rotten kingdom. This indicates the depths to which the nation has sunk since Claudius murdered his way to the throne. The whole atmosphere at court was so polluted by Claudius' moral corruption that there was really no way for Hamlet to survive in such a toxic environment without resorting to the same kind of underhand political tactics used by his hated stepfather.
In the political life of a nation, it's the ruler who sets the tone, and under Claudius, that tone was one of corruption, murder, and double-dealing at the very highest level. One of the many tragedies of the play is that fundamentally good people like Hamlet and Ophelia were unable to escape from this and ended up losing their lives due to Claudius' use and abuse of the power he stole.