"Tom Baker will never play the Dane."

Matthew Sweet offers a pleasingly Who aware piece in The Independent about the influences Hamlet may have had on the series:

"Dr Wiggins is equally crushing when I pitch him my theory that "The Invasion of Time" (1978) – in which the Doctor feigns madness to prevent his home planet falling to a race of aliens with thick Ulster accents – is another version of Shakespeare's story. If you must go truffling for Hamlet in pre-2005 Doctor Who, he suggests, better to try "The Masque of Mandragora" (1976), in which a fizzing alien entity arrives in Italy to stop the Renaissance from happening. It has an inexperienced, intellectual prince, a usurping duke, and a debate about the conflict between science and religion that recalls Hamlet's musings on the nature of the supernatural world."

Plus there was the original series bible for the 1996 revival in which Borusa was revealed to be the Doctor's father or something.