Born near St. Clairsville in Belmont County, Ohio, Tompkins moved with his parents to Morgan County in 1831 and settled near McConnelsville. He completed preparatory studies, and was graduated from the Ohio University at Athens in 1835. He studied law. He was admitted to the bar in 1837 and commenced practice in McConnelsville, Morgan County, Ohio. He served as recorder of McConnelsville in 1840. He served as prosecuting attorney of Morgan County 1848-1851. Street commissioner of McConnelsville in 1850. He served as member of the Republican State convention in 1855.
Tompkins was elected as a Republican to the Thirty-fifth and Thirty-sixth Congresses (March 4, 1857-March 3, 1861). He served as chairman of the Committee on Militia (Thirty-sixth Congress). He was an unsuccessful candidate for renomination in 1860. He resumed the practice of law. He died in McConnelsville, Ohio, July 23, 1862. He was interred in McConnelsville Cemetery.