Murray was the granddaughter of a slave and great-granddaughter of a slave owner. She would have been a life-long resident of Maryland, but accidents of fate intervened. Murray was one of six children and as a young child, her father developed a serious illness. When her mother died, the family was split up. Murray went to live with her maternal aunt and grandparents in Durham, North Carolina. Growing up in the "colored" section of Durham, she rebelled against the segregation that was an accepted fact of life  and refused to attend a segregated southern college. Upon applying to Hunter College in New York, she was both disheartened and motivated when she was rejected for admission. Determined to gain entry, she attended high school for a year in New York where she was the only Black among 4,000 students.

Ten years after leaving Durham and with a Hunter College degree, she fought for admission to the graduate school of the University of North Carolina - a school originally funded by her own slave-owning ancestors. Murray  was denied admission on the basis of race. She applied to Harvard Law School for graduate study, but was rejected because of her sex. She felt compelled to start a letter-writing campaign which put the Dean, the faculty, and the Harvard Corporation to shame. As she learned from personal experience, "One person plus one typewriter constitutes a movement."

Several years later, she entered Howard University Law School with the intention of destroying the Jim Crow laws, but came up against their cousin, the "Jane Crow regulations". In 1943, employing one of the earliest uses of non-violent tactics, she and a group of women students from Howard University successfully organized the first sit-in demonstration resulting in the desegregation of a cafeteria in Washington, D.C.

Murray's first book, "State's Laws on Race and Color", was published in 1951, and became an invaluable reference for civil rights lawyers. After publishing several books and poems she moved beyond published works. Her letter-writing campaign to the White House challenging the Roosevelt Administration on domestic policies led to a long-lasting friendship and professional relationship with Eleanor Roosevelt.

In many ways, Murray was an extraordinary woman. At the age of 62, she decided to become an Episcopal priest. In 1978, Murray was ordained in Washington, after a year of training as a special student at the General Theological Seminary in New York and after the Episcopal Church decided to admit women to the priesthood, . She became the first Black woman and one of the first women to be ordained in the Episcopal Church. 

In 1966, Murray was co-founded the National Organization for Women and served on the national board of the American Civil Liberties Union.