Jessup was raised in a series of orphanages in Georgia until he left at 16 to join the United States Merchant Marine. He was a merchant seaman and junior officer for 11 years during which time he said he read a book every day. He learned to be a writer by copying War and Peace on a typewriter whilst afloat, corrected all the errors, then threw the work over the side.

His first published novel was The Cunning and the Haunted published in 1954 based on his experiences in orphanages. In the same year Jessup wrote a teleplay for Tom Corbett, Space Cadet. The novel was filmed as The Young Don't Cry in 1957 with Jessup writing the screenplay for the film with Sal Mineo as the lead.

He began writing Westerns in 1957 with Cheyenne Saturday and finishing with Chuka where he wrote the screenplay for the film of the same name for actor and producer Rod Taylor. Jessup wrote a series of five Westerns featuring Wyoming Jones under the name Richard Telfair. With his Western series ending, in the same year he wrote again as Telfair for a series of spy novels featuring Montgomery Nash. He used the name Telfair for a novelisation of Danger Man called Target for Tonight in 1962.

Inspired by The Hustler, Jessup wrote a novel of poker playing called The Cincinnati Kid that was filmed with Steve McQueen. Another of his novels The Deadly Duo was also filmed.

In 1969 he wrote Sailor based on his experiences as a merchant seaman.

Otto Preminger bought the rights to his novel Foxway for filming, but the movie was never made.

His final work was Threat published in 1981.

He died of cancer.