Triplett was born in Franklin County, Kentucky. He served in the state House of Representatives in 1840 and in the state Senate from 1848 to 1852.
During the Civil War, he served as a major in the Confederate Army. He was elected and served as a representative from Kentucky to the Second Confederate Congress from 1864 to 1865. After the war he served as a state court judge. He was buried at Rosehill Elmwood Cemetery.