Alexander was born in Honolulu April 2, 1833. His father was missionary William Patterson Alexander and mother Mary Ann (McKinney) Alexander. He graduated from Punahou School in 1849 and Yale in 1858. He returned to Hawaii and joined the faculty of Punahou School as a professor of Greek and history. He became the fourth president of Punahou (then called Oahu College) in the summer of 1864. On November 6, 1874 he was appointed to the Board of education of the Kingdom of Hawaii, and then in 1896 Commissioner of Public Instruction under the Republic of Hawaii. He is most known for his "A Brief History of the Hawaiian People," an extensive history used by the educational system of the Kingdom and later the Republic.