The 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". 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".)
The Doctor mentions that he was a father ("The Doctor's Daughter"), a brother ("Smith and Jones") and a grandfather ("The Rings of Akhaten"), and he's referred to having a father ("Last of the Time Lords") and a human mother ("Doctor Who", the 1996 TV movie). As of 2014, the series has yet to fully explain the rationale behind the latter.
Jenny, from the TV story "The Doctor's Daughter", is sort of his daughter, having been cloned from his DNA (and the Doctor accepted her as such), and in a sense the Meta-Crisis Doctor could be considered a brother.
In contemporary sense, the Doctor is the last of his kind. His people are eternally locked in the Time War. He's said repeatedly that his family are all gone, but of course he could be wrong, for example, if Susan was left on Earth with a husband, it's possible she could have descendants unknown to the Doctor who have lived on.
The Doctor has mentioned marriage on a couple of occasions, such as in "Blink". It is known that the Doctor was once married to Queen Elizabeth I ("The Day of the Doctor"), and 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, which 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".
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.