Clennon was born in Waukegan, Illinois, the son of Virginia, a homemaker, and Cecil Clennon, an accountant. He attended the University of Notre Dame from 1962 to 1965. He studied at the Yale School of Drama for three years and became a member of their professional acting company. In 1996 he married Perry Adelman, a writer, camera assistant and photographer. They have twin children Daisy Virginia and Harry Francis.
He was a staunch opponent of the Vietnam War, often participating in protests, and remains politically active. In 2013, he repeatedly spoke out against the film Zero Dark Thirty and refused to vote for it for an Academy Award, stating that it promotes the acceptable use of torture.
I firmly believe that the film Zero Dark Thirty promotes the acceptance of the crime of torture, as a legitimate weapon in America’s so-called war on terror. In that belief, following my conscience, I will not vote for Zero Dark Thirty in any category…I cannot vote for a film that makes heroes of Americans who commit the crime of torture.