- You may be looking for the titular monarch.
The premiere night of the play was at the Rose theatre in London in 1592, with Shakespeare in the title role. When Rose Tyler distracted Shakespeare away from the play in an attempt to save him from the Shadeys, the Doctor took on the role of Richard III. The play was interrupted when rival playwright Robert Greene tried attacking the theatre with the Shadeys' powers, but the Doctor and Rose convinced Greene to save the world and banish the Shadeys to the dimension they came from instead. (COMIC: A Groatsworth of Wit)
The Fifth Doctor took Peri Brown and Erimem to what turned out to be a substandard performance of the play in 1597. Later while drinking with Shakespeare, the Doctor chastised him for not doing his research and presenting such a distorted account of the events of Richard's reign. Infuriated by the Doctor's insults and account of actual events which implicated the bloodline of his beloved Queen Elizabeth I, Shakespeare sneaked aboard the Doctor's TARDIS and travelled back to April 1483 in an attempt to influence actual events to more closely resemble his play. (AUDIO: The Kingmaker)
The Sixth Doctor later paraphrased and then quoted, "My kingdom for a horse," from Act V, Scene IV of the play. (AUDIO: The Hollows of Time) The Tenth Doctor did likewise in Goritania in the 1780s. Donna Noble did not recognise the quotation. (AUDIO: Death and the Queen)