Her father was named Amasa Coleman Lee. He was a lawyer, and the character of Atticus is loosely based on her dad.
Her mother was named Frances (Finch) Lee, hence the name "Finch" from the novel.
Lee studied Law in college, but never graduated. She worked as an airline reservation clerk in New York City, but left that job to write full time.
She never wrote another novel after Mockingbird. She wrote for columns for her various school publications, and has published two or three short papers. But there's nothing major from her since Mockingbird.
She is quite reclusive. The 40th Anniversary edition of Mockingbird boasts "With a new forward by the author!" Here is the text of that forward:
Please spare Mockingbird an Introduction. As a reader I loathe Introductions. To novels, I associate Introductions with long-gone authors and works that are being brought back into print after decades of interment. Although Mockingbird will be 33 this year, it has never been out of print and I am still alive, although very quiet. Introductions inhibit pleasure, they kill the joy of anticipation, they frustrate curiosity. The only good thing about Introductions is that in some cases they delay the dose to come. Mockingbird still says what it has to say; it has managed to survive the years without preamble. Harper Lee 12 February 1993It was never intended as an introduction - it was a plea, from a letter Lee wrote to her publisher, explaining why she was not going to write an Introduction!