Almost nothing is known about the Doctor's grandparents. The only reference to them in the entire Whoniverse can be found in "Cold Fusion" (Virgin New Adventures novel), where his grandfather is briefly referred to, but no information about him is given.
According to the Thirteenth Doctor in "It Takes You Away" the Doctor had seven grandmothers, their favorite being "Granny Five", Granny Five also claimed that Granny Two was a secret agent for the Zygons.
In "Doctor Who", the Eighth Doctor claims that his mother was human. Despite this, she seems to be knowledgeable about Gallifreyan culture, even singing him the Zagreus nursery rhyme ("Seasons of Fear" (Big Finish monthly audio story)). The mysterious woman who appears in "The End of Time" may or may not be the Doctor's mother. According to one source, the Doctors father was called Ullyses and his mother Penelope Gate.
In "Fear Her" the Doctor says 'I was a dad, once', presumably referring to Suasan's mother. Later, in "The Doctor's Daughter", a clone daughter is created from his DNA. In "Father Time" (BBC Eighth Doctor Adventures novel), the Doctor adopts a daughter Miranda Dawkins, whose biological father was another incarnation of the Doctor (known as the Emperor).
Th only full blood relative who has definitely appeared on screen is Susan, his granddaughter, who was his first companion and was the child of the title of the first episode, "An Unearthly Child". According to this first ever Doctor Who serial, she also invented the name TARDIS. She was played by Carole Ann Ford. (Note: some of the novels have suggested Susan was not in fact related to the Doctor, but to date no TV episode has disputed it, with Susan being referenced as the Doctor's granddaughter as recently as "The Rings of Akhaten" and "The Day of the Doctor".) John and Gillian are two young children who appear to be his grandchildren in various comic stories, starting with "The Klepton Parasites" (TV Comic comic story), but have been retconned in various ways by other stories; the novels "Conundrum" (Virgin New Adventures novel) and "Head Games" (Virgin New Adventures novel) theorize that they are merely creations of the Land of Fiction, while "The Land of Happy Endings" (Doctor Who Adventures comic story) suggests them to be nothing but dreams. Zezanne ("Sometime Never..." (BBC Eighth Doctor Adventures novel)) is the daughter of Miranda Dawkins, who is his (the Doctor's) adopted daughter.
The Doctor has mentioned marriage on a couple of occasions, such as in "Blink". Clara also claimed him to have been 'married four times, all deceased' when pretending to be him in "Death in Heaven". It is known that the Doctor was once married to Queen Elizabeth I ("The End of Time"), but he ran away immediately after the wedding ("The Day of the Doctor"), resulting in her wanting to kill him ("The Shakespeare Code"). At one point, he was also married to Marilyn Monroe ("A Christmas Carol"), although the Doctor claimed that the marriage didn't count. After numerous hints, in the 2011 episode "The Wedding of River Song", he married River Song. He was also presumably (although not necessarily) married to Suasan's grandmother. Mickey Smith teased Rose over the fact that the Doctor called Cleopatra 'Cleo' ("The Girl in the Fireplace"). Later, River Song confirmed the Doctor to have married Cleopatra at some point ("The Wedding of River Song"). The Eighth Doctor entered into a marriage of convenience with a brothel owner named Scarlette following the destruction of Gallifrey during the Second War in Heaven (a conflict that predated the Last Great Time War) ("The Adventuress of Henrietta Street" (BBC Eighth Doctor Adventures novel)). He also married Patience ("Cold Fusion" (Virgin New Adventures novel), "The Infinity Doctors" (BBC Past Doctor Adventures novel), "Cold Fusion" (Big Finish audio story)). The TARDIS has been metaphorically compared to a wife a few times, most notably in the title of the episode "The Doctor's Wife".
The Doctor's marriage to River Song makes Amy Pond and Rory Williams his in-laws (a point which is supported by the fact that they refer to him this way in "The Wedding of River Song"). By extension, this would make Amy and Rory's parents "grandparents-in-law" of the Doctor; to date we have seen Amy's parents in "The Big Bang" (after they were restored to reality), and Brian Williams, Rory's dad, appears on screen for the first time in "Dinosaurs on a Spaceship". By the time of the events of "The Doctor, the Widow and the Wardrobe", the Ponds consider him family enough to have him over for Christmas dinner.
The Doctor also references a brother in "Smith and Jones" and the expanded universe of novels and audio dramas features a character named Irving Braxiatel, who appears primarily in the Bernice Summerfield and Gallifrey series, who is the Doctor's brother. This does not, however, mean the Doctor's surname is Braxiatel.
The Thirteenth Doctor mentioned having had sisters in "Arachnids in the UK". Irving Braxiatel, the Doctor's brother, hinted to the Oracle of the Lost that he had lost either a daughter or a sister. "Tears of the Oracle" (Virgin Bernice Summerfield New Adventures novel)
The Doctor has 4 Great-Grandchildren through Susan Foreman (one biological, 3 adopted), Ian Campbell, David Campbell Junior ("Legacy of the Daleks" (BBC Eighth Doctor Adventures novel)), Alex Campbell ("An Earthly Child" (audio story)) and Barbara Campbell ("A Time & a Place" (Doctor Who Magazine short story))