He attended the public and private schools and graduated from Williams College (in Williamstown, Massachusetts) in 1869 and from Harvard Law School in 1871. He was admitted to the bar in 1872 and commenced practice in New York City with residence in Brooklyn.
He was a member from Kings County of the New York State Assembly in 1882 and 1883, and was Speaker in 1883. On January 13, 1882, he was injured in the same train accident in which State Senator Webster Wagner was killed.
He was New York State Comptroller from 1884 to 1887, and was Mayor of Brooklyn from 1888 to 1891. Chapin was elected as a Democrat to the Fifty-second United States Congress to fill the vacancy caused by the resignation of David A. Boody and served from November 3, 1891 to November 16, 1892, when he resigned.
Chapin served as railroad commissioner of New York State from 1892 to 1897, and continued the practice of law, He was also financially interested in various enterprises. He died while on a visit in Montreal, Canada in 1936, and was buried at Woodlawn Cemetery, Bronx. Chapin's grandson Hamilton Fish IV also was a U.S. Representative from New York.