Born in Greenwich, Massachusetts, Alvord completed preparatory studies and was graduated from Dartmouth College, Hanover, New Hampshire, in 1827. He studied law. He was admitted to the bar in 1830 and commenced the practice of his profession in Greenfield, Massachusetts. He served as member of the State house of representatives in 1837. He served in the State senate in 1838.
Alvord was elected as a Whig to the Twenty-sixth Congress and served from March 4, 1839, until his death in Greenfield, Massachusetts, on September 27, 1839, before the Congress assembled. He was interred in Federal Street Cemetery.