Harkness was educated at Lafayette College (1854–56), graduated from the University of Rochester (1858), and studied medicine in New York City. He served as a surgeon in the Union armies during part of the American Civil War. From 1862 to 1865 he was an "aid in astronomy" at the United States Naval Observatory and then, after service on the monitor USS Monadnock (1865–66), was employed in the Hydrographic Office.