Jessup was born on July 18, 1952, in Moscow, Idaho, to Clifford and Alvina Jessup and was raised in Seattle, Washington. He received his BFA in painting from the University of Washington in 1975 and his MFA in painting from the University of Iowa in 1979. He has had professorships at Ohio State University, Georgia State University, Cornell University, the Hartford Art School and the University of North Texas. He is currently the Professor of Drawing and Painting in the College of Visual Arts and Design at the University of North Texas in Denton, Texas.