Dunham's early life is unknown, as he first appears in the historical record in 1834.
In 1837 Dunham opened a machine shop on North Moore Street in New York City, later partnering with William Browning to found H.R Dunham and Co., an engineering firm specializing in the manufacture of steam engines.
Dunham and Co. continued operating with both partners until Browning left the venture in 1847.
Dunham likely continued in the business until 1856, the last year his products were listed in a New York directory.
[1] As of 1849, Dunham was part of the American Institute of the City of New York's committee on manufacturers.