A native of Bridport, Dorset, Whetham became a partner in the firm of S Whetham & Sons, flax and hemp manufacturers. The company had offices in Gracechurch Street in the City of London, and Whetham moved to the capital, where he became involved in the local politics of the City. In 1843 he was elected to the Common Council as a representative of Bridge Ward. He became deputy of the ward in 1865 and became an alderman in 1871. He served as Sheriff of London and Middlesex in 1873 - 1874, during which term he was knighted. He was Master of the Worshipful Company of Leathersellers in 1873 - 1874.
Whetham represented the City on the Metropolitan Board of Works in 1883 - 1884. He was also a member oif the Thames Conservancy Board 1883 - 1885 and Chairman of the London and Blackwall Railway Company 1870 - 1885.
Whetham married Sophia Maria Langley, and the couple had eight children. He died at his residence in Gordon Square after a long illness in 1885.
- "Death of Sir Charles Whetham". The Standard. 5 September 1885. p. 5.
- Alfred P. Beaven (1908). Chronological list of aldermen: 1801-1912. "The Aldermen of the City of London: Temp. Henry III - 1912". The Aldermen of the City of London: Temp. Henry III - 1912. pp. 141–156.
- The London Gazette: . 11 August 1874.