I’m working up a small project on William Shakespeare’s appearances in the Doctor Who universe. I think I now have a comprehensive list of them, which I will list below the cut, listed in three different orders. Still thinking about how I might put this together more creatively.

1) Shakespeare’s personal timeline
1572: Shakespeare is kidnapped by Daleks as a boy, and meets Troilus and Cressida on his journey home with the Doctor (Time of the Daleks; Apocrypha Bipedium)
1592: The alien Shadeys attempt to use Greene’s hatred of Shakespeare to invade Earth (A Groatsworth of Wit)
1593: Marlowe avoids death and disguises himself as Shakespeare (All Done with Mirrors)
1597: The Doctor puts Shakespeare right about Richard III, with unexpected consequences (The Kingmaker)
c. 1598: Queen Elizabeth gives Shakespeare the idea for The Merry Wives of Windsor, and Francis Bacon gives him the idea for Hamlet (The Chase)
1599: The alien Carrionites attempt to invade Earth using the text of Love’s Labour’s Won (The Shakespeare Code)
c. 1600: Sarah Jane Smith complains to Shakespeare about the sexism of The Taming of the Shrew, and suggests the character of Lady Macbeth to him (The Stranger, The Writer, His Wife and the Mixed Metaphor)
c. 1601: The Doctor writes Hamlet for Shakespeare, who has sprained his hand writing sonnets (City of Death)
c. later in 1601: Shakespeare is transported back briefly to 1505 where after an argument in Leonardo’s studio K9 destroys the original Hamlet manuscript
1609: Shakespeare visits Venice as a secret agent of James I and meets Marlowe again (Empire of Glass).
(There are a couple more vignettes from the beginning of Managra, in 1613, and the end of Empire of Glass, in 1616, but I’ll leave them for now.)

2) The Doctor’s personal timeline
First Doctor: Queen Elizabeth gives Shakespeare the idea for The Merry Wives of Windsor, and Francis Bacon gives him the idea for Hamlet; the Doctor meets Shakespeare and Marlowe in Venice
Fourth Doctor: writes Hamlet for Shakespeare, who has sprained his hand writing sonnets; Marlowe avoids death and disguises himself as Shakespeare; Sarah Jane Smith complains to Shakespeare about the sexism of The Taming of the Shrew, and suggests the character of Lady Macbeth to him; Shakespeare is transported back briefly to 1505 where after an argument in Leonardo’s studio K9 destroys the original Hamlet manuscript
Fifth Doctor: puts Shakespeare right about Richard III, with unexpected consequences
Sixth Doctor: says he must see Shakespeare again
Eighth Doctor: rescues the boy Shakespeare from kidnapping by Daleks, and meets Troilus and Cressida on their journey home
Ninth Doctor: prevents the alien Shadeys’ attempt to use Greene’s hatred of Shakespeare to invade Earth
Tenth Doctor: prevents the alien Carrionites’ attempt to invade Earth using the text of Love’s Labour’s Won.

3) order of broadcast/publication/release
1965: The Chase (TV, by Terry Nation): The First Doctor, Ian, Susan and Vicki watch as Queen Elizabeth gives Shakespeare the idea for The Merry Wives of Windsor, and Francis Bacon gives him the idea for Hamlet
1975: Planet of Evil (TV, by Louis Marks): The Fourth Doctor comments to Sarah Jane Smith that Shakespeare was a dreadful actor
1979: City of Death (TV, by "David Agnew", ie David Fisher, Douglas Adams and Graham Williams): reportedly, the (Fourth, presumably) Doctor wrote Hamlet for Shakespeare, who had sprained his hand writing sonnets
1985: Mark of the Rani (TV, by Pip and Jane Baker): The Sixth Doctor tells Peri that he must see Shakespeare again some time
1992: Doctor Who – City of Death (unofficial novelisation, by David Lawrence): Shakespeare is transported back briefly to 1505 by the Fourth Doctor, where after an argument in Leonardo’s studio K9 destroys the original Hamlet manuscript
1993: The Stranger, The Writer, His Wife and the Mixed Metaphor (short story published in Doctor Who Monthly’s Brief Encounters series, by Graham Cox): reportedly, Sarah Jane Smith complained to Shakespeare about the sexism of The Taming of the Shrew, and suggested the character of Lady Macbeth to him
1995: The Empire of Glass (Virgin Missing Adventures novel, by Andy Lane): The First Doctor, Vicki and Stephen meet Shakespeare and Marlowe in Venice
2002: Time of the Daleks (Big Finish audio play, by Justin Richards): The Eighth Doctor and Charley Pollard rescue the boy Shakespeare from kidnapping by Daleks.
2003: Apocrypha Bipedium (short story in Short Trips: Companions, by Ian Potter): The Eighth Doctor, Charley Pollard and the boy Shakespeare meet Vicki/Cressida and her husband Troilus
2004: All Done With Mirrors (short story in Short Trips: Past Tense, by Christopher Bav): Marlowe avoids death and disguises himself as Shakespeare, with help from the Fourth Doctor and Sarah Jane Smith
2005-06: A Groatsworth of Wit (comic strip published in Doctor Who Magazine, by Gareth Roberts): The Ninth Doctor and Rose prevent the alien Shadeys’ attempt to use Greene’s hatred of Shakespeare to invade Earth
2006: The Kingmaker (Big Finish audio play, by Nev Fountain): Peri and Erimem get caught up in the unexpected consequences of the Fifth Doctor’s attempt to put Shakespeare right about Richard III
2007: The Shakespeare Code (TV, by Gareth Roberts): The Tenth Doctor and Martha prevent the alien Carrionites’ attempt to invade Earth using the text of Love’s Labour’s Won.

For my money, the best are The Shakespeare Code, which is also the most recent; The Kingmaker, which of course I have described here without spoilers as it violates Shakespeare’s own continuity so badly; and the short story Apocrypha Bipedium. On the other hand, I would not especially recommend that you bother seeking out either Time of the Daleks or All Done with Mirrors.