As we are limited in space, below are a few ideas to help get you started.
Cultural criticism and new historical criticism are actually closely related. The difference between the two concerns points of emphasis. Cultural criticism specifically emphasizes analyzing literary works that have been "marginalized by the aesthetic ideology of white European males" (Michael Delahoyde, Introduction to Literature, "Cultural Criticism"). In other words, cultural criticism is used to analyze literary works that have been belittled by globally dominant white European males because the works contradict what white European males classically believe to be beautiful in art and literature. Hence, analyzing literature in terms of what has been classically marginalized by white European males also becomes a process of investigating in what ways the privileged "race, class, and gender" has influenced art and literature. Cultural criticism can especially be used to analyze any literature or works of art written and produced by "minority ethnic groups and postcolonial writers" ("Cultural Criticism"). It will look at any works pertaining to "folk, urban, and mass culture," such as popular fiction and even soap operas ("Cultural Criticism").
Since Graham Greene's The Comedians can be classified as popular fiction, particularly a political thriller, we certainly can use cultural criticism to analyze the book. However, what's also interesting is that the author himself is not an ethnic minority; Greene was an Englishman who worked as a journalist, film and book reviewer, novelist, and who was eventually recruited by MI6. It was while working for MI6 that Greene eventually traveled to Haiti in 1954, staying for only a short time ("Graham Greene"). Greene stayed in the Hotel Oloffson in Port-au-Prince, which because the hotel setting for the novel, but he renamed it the Hotel Trianon ("Graham Greene"; "The Comedians"). Haiti was at this time also ruled by Francois Duvalier, also called Papa Doc, whose dictatorship significantly influenced the novel ("Graham Greene"). Therefore, while this novel is a popular novel, it should also be noted that it was written by a member of the white European male class, the same class that influences art and literature. Hence, as you analyze the novel using cultural criticism, you can pay attention to details like how Greene portrays the suffering minorities he is writing about. Can any stereotypes be seen? Does his characterizations of minorities seem objective or subjective? You can also think about Greene's characterization of Papa Doc. Does that characterization fit the view of those he ruled over? Or is Greene's characterization influenced more by the political and cultural ideals of white European males? Essentially, as you analyze the book through cultural criticism, you want to explore just how accurately portrayed the marginalized minority culture that influenced the novel.