Why is it that Hamlet's uncle killed his father in Shakespeare's Hamlet?

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-

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our sometime sister, now our
queen,

and of his involvement
in the political affairs of Denmark-

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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.