After studying at the Medway College of Art from 1959-61, he obtained a scholarship from the Royal College of Art to study print-making and gained a first and a silver medal for works of distinction. On leaving college in 1964, he took part-time posts teaching at St. Martin's College of Art and Maidstone College of Art. In 1970 his teaching activities took him to Wolverhampton Polytechnic where he was a regular visiting lecturer.

He occasionally worked as a master printer in lithography for other artists and also at Sky Editions for Christie's Contemporary Art. In 1982 he completed a series of prints he had worked on with Jim Dine. Bessant's work has been widely exhibited in the UK and is also in private collections in Europe and the USA.