His father was a computer-systems designer with Hiram Walker Inc.[3] Bissell was an athlete who enjoyed performing feats of daring: at the age of 8 he jumped off a 30-foot (9.1 m)-high diving board, even though he did not know how to swim.

[4] He was noticed by the American ballet dancer Edward Villella, who encouraged his parents to send him to a performing arts boarding school.

[4] Bissell then spent a year at the North Carolina School of the Arts which he left when he was informed that he should pay more attention to his academic studies.

He then won a scholarship to study at the School of American Ballet, where he was encouraged by Lincoln Kirstein, its founder, and Stanley Williams, one of his teachers.

In 1978, he was promoted to soloist and to principal dancer in 1979 at the American Ballet Theatre due to the shortage of men in the company—even making the cover of Dance Magazine.

The ballerinas nicknamed him "Tarzan," as he was a huge, hulking juggernaut of a man who could carry some of the biggest and tallest girls in the company.

[5] Bissell and Kirkland then appeared as guest artists with the Eglevsky Ballet in its production of Act II of Giselle in 1982 at the Hofstra Playhouse in Hempstead, Long Island, New York.

He appeared in many lead roles, including Don Jose in Roland Petit's Carmen, Franz in Coppélia, Basil and Espada in Don Quixote, Albrecht in Giselle, Romeo in Romeo and Juliet, Prince Siegfried in Swan Lake, James in La Sylphide, Prince Desire in Sir Kenneth MacMillan's Sleeping Beauty and lead roles in George Balanchine's Stravinsky Violin Concerto, Symphonie Concertante and Theme and Variations.

Bissell started using drugs at a very young age, including shooting up with heroin, and consumed large amounts throughout the course of his life.

However, a plea bargain was made whereby the judge ordered him to arrange to give a performance at Indiana University with the proceeds to be given to charity.

[11] In 1984, company officials from the American Ballet Theatre consulted with experts on drug addiction and found a therapist for him.

[1] Prior to entering the clinic he had injured his foot and was thus prevented from going on the American Ballet Theatre's fall tour.

At the time of his death, he was engaged to fellow dancer at the American Ballet Theatre, soloist Amy Rose, and had planned to rejoin the company the following month.

[1] His death prompted charges of extensive drug use in the dance world by Bissell's parents and fellow-dancer Gelsey Kirkland.

[4] Kirkland's autobiography Dancing on My Grave mentions Bissell's frequent use of cocaine[4] and, when discussing her own addiction, she alleged that he had introduced her to the drug.

Gelsey Kirkland alleged that Bissell's death was "an unavoidable tragedy caused at least in part by the failure of the ballet world and American Ballet Theater in particular to acknowledge and deal openly with the drug problem",[12] which contrasted with Dillingham's statement that "his death came as an utterly horrible surprise".