Beaman was born in Chester, Vermont, and moved with his parents to a farm in Franklin County, New York, in 1819. He attended the district schools and Malone Academy in Malone, New York. He taught school and moved to Rochester, New York, in 1836, where he studied law. He moved to Manchester, Michigan, in 1838, where he was admitted to the bar and commenced practice in 1839. He moved to Tecumseh, Michigan in 1841 and practiced law there and in Clinton. He moved to Adrian in 1843, having been appointed prosecuting attorney for Lenawee County, and he served in that position until 1850. He was the city attorney of Adrian and a member of the convention that organized the Republican Party "under the oaks" at Jackson in 1854. He was a delegate to the first Republican National Convention at Philadelphia in 1856. He was also mayor of Adrian in 1856 and judge of the probate court of Lenawee County 1856-1860.

Beaman was elected as a Republican from Michigan's 2nd congressional district to the United States House of Representatives for the Thirty-seventh and to the four succeeding Congresses, serving from March 4, 1861 to March 3, 1871. Due to redisctricting after the 1860 U.S. Census, Beaman represented Michigan's 1st congressional district following the elections of 1862. He was Chairman of the House Committee on Roads and Canals in the Thirty-ninth Congress. Beaman declined to be a candidate for renomination in 1870 and returned to Adrian where he resumed the practice of law. He was again appointed judge of probate of Lenawee County in 1871, was then elected to the same position in 1872, and re-elected in 1876.

Due to ill health, Beaman was declined appointment as United States Senator to fill the vacancy caused by the death of Zachariah Chandler in 1879. He also declined appointments to the Michigan Supreme Court and as United States Commissioner of Indian Affairs. He died in Adrian and is interred in Oakwood Cemetery there.