As a former HS history teacher that took some significant time to enhance my own knowledge in this area (I got my World History teaching cert with no requirements to study any history outside of Europe and the US), I highly highly HIGHLY recommend John Reader's book Africa: Biography of a Continent. It starts with geologic history, then evolutionary, then tribal, then ancient African societies, then middle and modern societies. It has excellent chapters on Axum, Jenne-Jeno, and others. It's a great survey and includes context (which is especially important) and discussions of evidence, all presented in an interesting yet rigorous way.

I read the book "Warfare in Atlantic Africa, 1500-1800" based on the AskHistorians reading list. It had a good summary of the political situation in West Africa as well as the different military tactics that were implemented (hence the title of the book). A lot of it is based on records kept by Europeans during the Exploration and Slave Trade eras, so it might not "ancient" enough for your interests.

Still, I think reading this book did give me a decent picture of the political structure of West Africa before it was colonized. Also, if you google the title you can find a free PDF of it.