Born in Chester County, Pennsylvania, Maxwell moved with his mother to Virginia about 1800. He completed a preparatory course. He studied law. He was admitted to the bar and commenced practice in Weston, Virginia (now West Virginia). He served as member of the State house of delegates in 1821–1824.
Maxwell was elected as an Adams candidate to the Twentieth Congress. He was reelected as an Anti-Jacksonian to the Twenty-first and Twenty-second Congresses (March 4, 1827 – March 3, 1833). He served as chairman of the Committee on Expenditures in the Department of War (Twenty-first Congress), Committee on Expenditures in the Department of the Navy (Twenty-second Congress). He was not a candidate for renomination in 1832. He resumed the practice of law and was also engaged as a surveyor and land patentee. He died in West Union, Virginia (now West Virginia), February 13, 1862. He was interred in Odd Fellows Cemetery.
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