Clover Margaret Collins was born in 1945 and grew up in the suburb of Gordon, on Sydney's North Shore, one of three daughters of Kathleen and Francis Collins. She attended Loreto Kirribilli at Kirribilli and Elm Court Dominican Convent, Moss Vale. Moore matriculated to the University of Sydney, and obtained a Bachelor of Arts in 1969 and a Diploma of Education from the Sydney Teachers' College, while residing at Sancta Sophia College. After graduation she began work as an English and History teacher at St Ives High School and Fort Street High School, before moving to London to teach for several years. Moore married Peter Moore, an architect, in 1972 and they had two children, Sophie and Tom. They returned to Australia five years later and settled in the inner-city suburb of Redfern. As a young mother in the Labor Party-dominated South Sydney Municipal Council, Moore became involved in a local resident action group and decided to run for Council in 1980 when she and other members of the group met, after three years of attempts, with the Mayor, Bill Hartup, regarding a local park which Hartup had demanded to have its grass replaced with asphalt (to aid street-sweepers in seeing broken glass), surrounded by barbed wire (to keep out the drunks at night), and to have its lone tree removed (a nuisance).