Born in Mount Carmel, Mississippi, Graves attended Centre College in Danville, Kentucky.

At the outbreak of the Civil War, he joined the Confederate States Army and served under General Nathan Bedford Forrest and was paroled with him at Gainesville, Alabama, in May 1865.

After being mustered out, he returned to college and was graduated from Oakland (later Alcorn) University, Mississippi, in July 1867.

He was admitted to the bar and practiced law in Lexington, Missouri where he was city attorney in 1872.

The Alexander and Elizabeth Aull Graves House was listed on the National Register of Historic Places in 1993.