After graduating from Harvard University in 1962, he served in the U.S. Air Force Reserves, and worked as an editorial writer for the New York Herald Tribune. In 1966 Chapman moved to Seattle and wrote a book entitled The Wrong Man in Uniform, arguing against conscription.

With his college roommate George Gilder, he wrote an attack on the anti-intellectual policies of Barry Goldwater entitled The Party That Lost Its Head (1966).

Chapman became active in politics through the Seattle Young Republicans, and became a member of the United States Republican Party. He was elected to the Seattle City Council in 1971. In 1975, he was appointed Secretary of State of Washington. He campaigned for the office of Governor of Washington in 1980, but ultimately did not win the Republican nomination.

Chapman was appointed by President Ronald Reagan to the position of Director of the United States Census Bureau and served in that role from 1981 until 1983. Between 1983 and 1985 he was Deputy Assistant to President Reagan and Director of the White House Office of Planning and Evaluation. From 1985 to 1988 he served in the appointed position of United States Ambassador to the United Nations Organizations in Vienna. His portfolio included nuclear proliferation, refugees, economic development, and the control of narcotics.

From 1988 to 1990 Chapman was a fellow at the Hudson Institute, a conservative think tank. In 1990 , he left Hudson and founded the Discovery Institute. The institute is best known as the hub of the Intelligent Design movement, and also focuses on a broad range of issues, including transportation and international cooperation in the Cascadia region.

Chapman is a Roman Catholic and is married to the former Sarah Gilmore Williams (a great-granddaughter of Theodore Roosevelt), with whom he has two sons, Adam and Andrew.