There are two possible motives for Claudius killing King Hamlet (Hamlet's father). The first would be to access to the throne of Denmark; the second to take Gertrude, the queen, for his wife. It is likely that Claudius' passion for both prizes was his driving force. His greatest folly was perhaps the speed at which Claudius took on the accession to both roles, as he says himself in his primary speech -
Though yet of Hamlet our dear brother's death
The memory be green,
As can be seen in Claudius' detailed instructions in Act 1 Scene II, he has usurped the role of king fully, in terms of his marriage to Gertrude-
our sometime sister, now our queen,
and of his involvement in the political affairs of Denmark-
To our most valiant brother. So much for him.
Now for ourself, and for this time of meeting.
Thus much the business is:
Claudius' speed and enthusiasm for each of the roles he has taken by the murderer of King Hamlet make both the spirit of the dead king, and his tormented son, seek revenge.