Dickerson was born in Urbana, Ohio to Amanda and Lee Dickerson, but she made Summit, New Jersey her home where she lived for more than 78 years. Her mother died when she was a teenager, and she lived with her father until she graduated from high school in 1930. She lived with relatives including her grandfather, Benjamin, an ex-slave, who had witnessed a Civil War battle at the age of ten in the state of Virginia. She was denied entrance to the Diller-Quaile School because of racial discrimination but studied piano with Helen Chrystal Bender of the Summit School of Music. She worked as a domestic, and at one point, she worked at Bell Labs before becoming a music teacher. She attended Columbia University and New York University and graduated from the Juilliard School in New York City. She was a member of the Wallace Chapel A.M.E. Zion Church for 80 years and served as the church's historian.
Dickerson taught piano to thousands of students over a period of seventy-eight years. She emphasized precision, sticking strictly to piano compositions as written, and practice; in 2004, she was quoted in the Los Angeles Times for stressing the benefits of practice. She taught New Jersey preschoolers in Cranford and Millburn and she taught music to the hearing impaired. According to one estimate, her students live in all fifty states in the United States.
Dickerson taught some notable students. She taught local resident Lawton C. Johnson of Summit who became a church organist as well as a locally recognized educational administrator; in 2004, the Summit Board of Education decided to rename the middle school after him. She taught Graeme Cowen and composer Carolyn Schmidt, who became director of the musical group named the Hickory Tree Singers. She taught piano to New Jersey's French hornist Tom Varner who formed the Tom Varner Quartet. Varner wrote "I loved my teacher Ms. Capitola Dickerson–she would play 78s of Art Tatum for me."
Dickerson loved teaching children, including kindergartners and preschool children. She taught music to hearing impaired children at the Summit Speech School for thirty years. She taught preschoolers at the Westfield Day Care Center from 1978 to 2000, teaching rhythm, movement, songs and music appreciation to thousands of children. Sally Rogers, one of her pupils, wrote:
"I remember singing in kindergarten and how important that was. And I also remember a woman in our town named Capitola Dickerson. I went to one class she did at the Y where she was sharing rhythm instruments with us. When she handed me a tambourine, her manner communicated kindness and inclusion. It may seem hard to believe because of how young I was, but she changed my life. I actually sought her out ten years ago and called her up and put her on the CMN quilt. She respected me, shy kid that I was, and modeled how to invite children to come into music their own way."
Dickerson was regularly honored by the city of Summit. She played on Martin Luther King day in 2008. She played piano and sang along with the protégé performers in honor of one of the Tuskegee airmen George Watson, in a program of remembrance, sponsored by Summit Supports Our Troops. She received a Keeper of the Dream Award from the city of Summit, New Jersey. In 2011, she was honored by the City of Summit; in a speech before the Common Council, Mayor Glatt described how Dickerson, who was in a shelter during a power outage, made a phone call from the shelter to ask if her neighbors needed help. Mayor Glatt said:
...as soon as she sat down, she pulled out her phone to call her neighbors to see if they had heat or needed help. This type of concern for others is typical of Dickerson, who has lived in Summit for more than 70 years.
Dickerson's contributions have been honored by various organizations including the Summit Area YMCA, The Connection for Women and Children, the United Way, the Tri-City Branch of the NAACP, the Summit Chamber of Commerce, Church Women United, Soroptimist International, Links, The Boy Scouts, The A.M.E. Zion Church, and the Wallace Chapel A.M.E. Zion Church.
Dickerson was described as a strong leader in efforts to promote affordable housing in Summit in cooperation with the first Sponsors Committee. She led by example and attended many committee meetings and active demonstrations. In mid-century, she was one of the first African-American women employed at Bell Laboratories. She encouraged women to "develop and use their gifts and talents in every aspect of society," according to a report in the Star-Ledger. She worked with Habitat for Humanity.