In Hamlet, how does Claudius's hunger for power result in him dying?
Well, you could argue that the means by which Claudius gains power is directly related to the way that he dies. It is of course his assassination of his brother, the rightful king, that gains him the crown, and he directly expresses in his soliloquy that he will face the consequences of his actions:
O, my offence is rank it smells to heaven;
It hath the primal eldest curse upon't,
A brother's murder. Pray can I not,
Though inclination be as sharp as will:
My stronger guilt defeats my strong intent...
There is a definite sense in which the fate of Claudius is certain from the moment that he kills his brother and assumes the position of king. Claudius in this speech clearly indicates the way in which he believes heaven will not let him rest unpunished because of the grevious nature of his crime, and thus his death is only a matter of time. His death therefore can be seen as a direct consequence of his action of killing his brother in the first place, as Claudius in his soliloquy expresses his belief that such a massive crime will not be allowed to go unpunished.