Ankomah attended Adabraka Presbyterian Junior High before continuing her education at Labone Senior High School. She then attended college at Jayee University College, where she studied marketing. Ankomah characterized her childhood as difficult; her mother struggled to support the family as a single mother and expected Ankomah's support as the first born. Ankomah attributes the challenges she faced as important opportunities that allowed her to learn and succeed.

Ankomah was introduced to acting by her cousin, Harry Amonoo. In 1994, Amonoo took her to the National Theatre where she joined the Harrisco Drama Group. She then began acting in stage dramas at Funworld and Kiddafest. While at Labone Senior High School, Ankomah joined the Theatre Programme for Schools (TPS), and participated in a number of plays. She subsequently started acting in television series. These included Suncity, St. James Hotel, All That Glitters and Where is Your Mobile?

She also auditioned for a number of movie roles. Ankomah placed third at the 2007 edition of the Next Movie Star. After several rounds of eliminations, she was left in the competition with five others: Lydia Forson, Uti Nwachukwu, Enoch Hammond, Ameh Lucy Franscesca and Oshinuga Olufemi. The winner was decided via votes from the public, views of professionals about the housemates, and an in-house judges' assessment. She was noted for her versatility and good acting skills: audibility, gesticulation, projection and mimicking. Martha Ankomah believes being in a competition is not enough. In an interview with mygong.net, the actress said coming out of a reality show does not make one a star.

“Next Movie Star didn’t do much for me, but you get to meet different kinds of people, and also learn from the directors who came to train us. When you come from Next Movie Star, it does not make you a star. You still have to go for auditions. That’s what people don’t realize. Do not think that because you have been on TV for 30 days you are there, forget it! It’s just the grace of God,” she stated.
Ankomah owns an ultra-modern beauty salon called ‘Martha’s Place’. The salon, located at Tesano, Maa Bee Plaza, a storey building adjacent Karldorlf was officially opened in 2013. It offers bridal makeup services, pedicures and manicures, facial treatments, acrylic waxing, barbering, and cosmetics.Ankomah was made the face of Vita Milk’s Viora in 2014. She shot a commercial for the product in early 2014 in Thailand.