Camp was born in the city of New Britain, Connecticut, the son of Leverett Lee and Ellen Sophia (Cornwell) Camp. He attended Hopkins Grammar School in New Haven, entered Yale College in 1876, and graduated in 1880. At Yale he was a member of Delta Kappa Epsilon fraternity. On June 30, 1888, Camp married Alice Graham Sumner, sister of William Graham Sumner. They had two children: Walter Camp, Jr. (born 1891) who attended Yale as well and was elected as a member of Scroll and Key in 1912, and Janet Camp Troxell (born 1897).