Grover was born February 18, 1819 near Phelps, New York where he attended the common schools before moving to Kentucky in 1837. He attended Centre College in Danville, Kentucky and taught school in Woodford and Franklin Counties. In addition, he studied law and was admitted to the bar in 1843 and commenced practice in Owenton, Kentucky.
Grover was a member of the Kentucky Senate 1857-1865 and a member of the Democratic State convention in 1863. He was elected as a Democrat to the Fortieth Congress (March 4, 1867 – March 3, 1869). Due to charges of disloyalty which were found to be unsustained, he was not seated until December 3, 1867. After leaving Congress, he resumed the practice of law. He moved to Georgetown, Kentucky in 1881 and continued the practice of law until his death in that city on July 20, 1887. He was buried in Georgetown Cemetery.