A brief list of obvious, and not so obvious places to check out if you are starting out on an African history dissertation/thesis:
National Archives, Kew: particularly the FO and CO series among others.
Birmingham Library Special Collections: Church Missionary Society Archive covering a wide range of African and other locations; good for missionary letters, diaries, mission station lists etc
Bodleian Library of Commonwealth and African Studies; excellent selection of books, personal papers etc
Durham Sudan Archive; fantastic resource for those studying Sudan, particularly during the colonial period.
Wellcome Centre, London: useful for medical history; particularly with relation to sleeping sickness in Africa
London School of Hygiene and Tropical Medicine: similar to above
Royal Geographical Society, London: collections containing personal papers, publications and maps of numerous explorers across Africa
Brotherton Library, Leeds: CMS microfilm collection + Correspondence, papers and publications of Professor Adrian Hastings.
Mill Hill Fathers, Freshfield, Liverpool: Missionary archive
London Universities: SOAS, LSE; apart from a superb range of works also good special collections