The following is all I could gather about Leslie Graves from what I found on the internet. Her first regular role was on the sitcom Here We Go Again. It looks like it was an interesting and contemporary type of show for the time for its depiction of divorced families. Here is the intro.
...And also a still which features Leslie (probably around 12-13 years of age).
Leslie Graves had a knack for portraying a sassy toughness on screen while still appearing adorable. Her father, Michael Graves, was a theatre actor and introduced her to the entertainment industry when she was around 5 years old. One of her earliest roles by age 10 was playing opposite Anne Bancroft in a Broadway play A Cry of Players (1968–1969) written by William Gibson. She then quickly moved into acting for television. As a child, she appeared on the first season of Sesame Street (1969, first 13 episodes) and had a brief appearance as a tough 11 year old babysitter on the The Mary Tyler Moore Show(1971, in the episode titled "Baby Sit-Com"). A couple of years later, she had a regular gig on a sitcom as part of a divorced family in the short -lived program Here We Go Again which starred a young Larry Hagman.(1973). Around this time, she was also featured in a variety of commercials.
In the late-70s, as a teenager, she left Hollywood. At first she lived on Mustang Island off the Gold Coast. Later, she supposedly moved with a boyfriend to Texas, where she worked on a shrimp boat for three years.
Somehow this led to CBS casting her, at 23, in the role of Brenda Clegg in the daytime soap Capitol. Her character was popular and Leslie played the role for two years.
On the set she found a sort of second, supportive mother in her co-star Carolyn Jones. When Jones died, Graves was devastated and suffered from severe depression. Also, around this time, Graves began to have a serious drug problem. In late summer 1984, it is rumored that Graves left the CBS show due to a a heroin overdose, although her departure was on record as being due to stress.