Since Robert Penn Warren won the Pulitzer Prize for this novel in 1947, it has to be literary fiction. You can stick anything in this genre that pays particular attention to character development, form, style, and usually difficult social issues. While plot is important, it is less important than the characters themselves, and the social issues they confront.
The family drama aspect of this tale peaks in Chapters Eight, Nine, and Ten. Family secrets are exposed, families torn apart, and new families are created. All the King's Men explores not only what family means to the characters in the novel, but to America before and after the Civil War.
Which brings us to the historical fiction aspect of the novel. Interestingly, in his Paris Review interview Warren denies that he writes historical fiction. Perhaps what he's getting at is that all fiction set in a real time period is historical fiction. If we pigeonhole a story as historical, we might lose sight of the larger issues, which stretch beyond history.