Costos was born in 1963 and grew up in Lowell, Massachusetts. He is a second-generation Greek-American, whose father served as a U.S. Marine and was stationed at Camp David during the Truman administration. He is the first in his family to graduate college. He earned a degree in political science from the University of Massachusetts in 1985.

In October 2013 the Spanish government summoned him to address allegations that the National Security Agency had recently collected data on 60 million telephone calls in Spain.

While visiting California in 2014, President Obama and his wife Michelle stayed at the Costos-Smith home.

In June 2015, he joined other gay U.S. ambassadors in a statement supporting international trade agreements, linking open markets to the development of open societies that provide civil rights protections.

After leaving the White House in January 2017, the Obamas departed Washington for Palm Springs, where they were guests in the Costos-Smith home for several days before traveling to the British Virgin Islands to stay at the home of Sir Richard Branson.

Costos lives in Los Angeles, California, with his partner Michael S. Smith. He is a vegetarian, but imposed no dietary restrictions on embassy functions. On the subject of LGBT rights, Costos has said "I am not an Activist. I broadly support human rights for straight, gay, women’s issues. And I just happen to be gay.... This doesn't define me as a person, but it is part of who I am. It's like being a vegetarian: it's one more facet of James Costos." He said he and Smith had been welcomed in Spain "with great accommodation" and that because Spain already provides LGBT civil rights, the role he and Smith played was a symbolic for those who still suffered from a lack of enforcement in support of their rights: "What we can do through our messaging is give a sense of hope."