We don't know all of the details, but we do know some of it.
- Susan Foreman, a companion to the First Doctor in his first 2 seasons, was the Doctor's granddaughter.
- the Doctor had parents, who may have been "The Woman" (a Time Lord that appeared in the End of Time Part 1 and 2) and the man seen with her, although that's not confirmed.
- the Doctor had at least one brother.
- the Doctor had children (one of whom was presumably Susan's mother or father).
- the Doctor had been married at least once.
In the novels:
- Irving Braxiatel may have been the Doctor's brother.
- the Doctor actually doesn't have parents, or children--in fact, no Gallifreyans from the Time Lord era do. Instead, he was born from a Loom into a House of Cousins, Lungbarrow. The novel of the same name gives the names of many of his 44 cousins.
- the Doctor was a reincarnation of The Other, who was born before the Time Lord era and presumably did have parents. The Other also had 13 children, including one of Susan's parents.
- the Doctor had a wife, Patience (who was also, earlier, the wife of The Other and mother of his 13 children).