Acheampong was born to Catholic parents of Ashanti origin. He attended the Roman Catholic schools at Trabuom and the St Peter's school (also Catholic) at Kumasi, both in the Ashanti Region of Ghana. He attended the then Central College of Commerce at Agona Swedru in the Central Region of Ghana. He was commissioned in the Ghana Army in 1959, and served as a member of the UN peacekeepers during the Congo Crisis.