Given the time and space constraints, I'll just focus on Achebe's most famous and influential novel, 1958's Things Fall Apart. Post-colonialism can be a nebulous and confusing term, as it applies both to a specific period in literature, as well as a certain thematic/ideological approach. On Oxford Biographies, it's defined as "a body of thought primarily concerned with accounting for the political, aesthetic, economic, historical, and social impact of European colonial rule around the world in the 18th through the 20th century."
Achebe was a Nigerian writer, and his country had been colonized by the British in the 19th century, eventually gaining its independence in 1960. So Achebe was writing at the tail end of the colonial era. In Things Fall Apart, he has, I think, two main purposes. One is to reclaim the cultural heritage of the country, specifically the Igbo people, which the British had destroyed through Westernization (religion, education, law). An oft-repeated colonial argument is that the Western powers brought "civilization" (a highly contentious term) to Africa. Achebe recreates the rich, diverse culture that existed before colonial rule.
Up to the point that Achebe and others began writing, their stories were told by the colonial powers, so they were represented but could not represent themselves. Therefore, Achebe's second main purpose is to tell a story that has not been told before; he wrote it in English so it would be accessible to a wide readership. Achebe details the negative, racist, and destructive effects of the Europeans on the indigenous peoples, and his novel would inspire a generation of writers to do the same.
The two sites below are useful resources and, for further reading, I'd suggest The Empire Writes Back and Said's Orientalism.