Born in the United Kingdom in 1815, Dunman came to Singapore in 1840 as an assistant in the merchant firm Dyce & Co. He entered the police force in 1843. He was made Superintendent of Police in 1851, and Commissioner of Police in 1856.
During his time heading the police force, Dunman was known for being on good terms with the people of various classes and communities within Singapore, and thus able to gain assistance and first-hand information regarding what was happening in the city. He was respected by leaders of the European community, and supported by influential Malays and Indians, who often felt powerless to prevent Chinese gangs from roving into their districts, assaulting people, and robbing homes and businesses. He improved the efficiency and training of the police force. Among the measures he introduced were improved pay and working hours for policemen, setting up training programmes and night classes for members of the force, and creating a pension scheme for retired policemen. Morale in the force improved and the crime rate in Singapore decreased under his leadership.
Dunman retired from the police force in 1871, and spent the next few years on his coconut plantation, Grove Estate (in what is now the Mountbatten area of Katong).
Dunman was the founding president of the Tanglin Club in 1865.
Dunman was also one of the founding members of Orchard Road Presbyterian Church in Singapore.