Whether Susan was actually The Doctor's grandaughter or not is a matter of much debate. But there is no doubt that there is a close bond between them. There is never any suggestion during the time we see them travel together that is not the case. Susan's name is probably not Foreman, any more than the Doctor's is. This is merely the name on the gates of the junkyard where the TARDIS is hidden. Susan looks and behaves like a typical 15-year-old girl. It is at here insistence that she attends Coal Hill School, and it is here that her strange breadth of knowledge is noticed by her teachers.  Ian Chesterton says, "She lets her knowledge out a bit at a time so as not to embarrass me...She knows more science than I will ever know. She's a genius." Yet as the history teacher, Barbara points out, her knowledge is patchy: "I don't know how you can explain a fifteen year old girl doesn't know how many shillings there are in a pound...She said she thought we were on decimal system." but Susan is not human. While she likes pop music, she can read a thick textbook in an evening and, as we later discover, under certain circumstancs she can communicate telepathically.(The Sensorites)
 
Susan is an adventurer. " I like walking through the dark," she tells her teachers. It is mysterious." In The Daleks, she braves the dangers of the petrified jungle and the mutated Thals to retrieve the drugs her grandfather and friends dsperately need. In The Aztecs, she is headstrong enough to refuse to marry the Perfect Victim who can demand anything of anyone on pain of death. 
When we first meet Susan, she is enjoying life as a 'normal' schoolgirl on twentieth century Earth. "I love your school," she tells Ian. "I love England in the twentieth century. but Susan is growing up fast. She may have a bit of a crush on Ian, and see Barbara as a mature confidante, but she falls completely in love with David Campbell (The Dalek Invasion of Earth). Realising that his grandaughter will never willingly leave him, believing him to be dependant on her, the Doctor locks her out of the TARDIS.