He played college football for the Temple Owls and was a member of the Jets' famed "New York Sack Exchange".

[3] Klecko led the Temple Owls in tackles his last three seasons (1974–1976), twice making the All-East team and receiving All-American mention as a junior and senior.

He was the ECAC's Rookie of the Week on October 27, 1973, after Temple's 31–8 victory over the University of Delaware Blue Hens, in which he had five sacks and 15 tackles.

In November 1981, Klecko, Gastineau, Salaam and Lyons were invited to ring the ceremonial opening bell at the New York Stock Exchange, which served as the inspiration for their nickname.

Hall of Fame center Dwight Stephenson, in describing Klecko as a "great defensive lineman", considered him one of the two best interior linemen he had ever faced.

[11] Hall of Fame tackle Anthony Muñoz said about Klecko, "In my 13 seasons, Joe is right there at the top of the defensive ends I had to block, up there with Fred Dean, Lee Roy Selmon and Bruce Smith.

"[12] Hall of Fame guard Joe DeLamielleure added that, "You can't think of his ten year period without him.

On August 16, 2010, during halftime of the New York Giants–Jets preseason game at the New Meadowlands Stadium, Klecko was inducted as a member of the inaugural class into the Jets' Ring of Honor.

In 2016, the Professional Football Researchers Association named Klecko to the PFRA Hall of Very Good Class of 2016.

[13] On August 17, 2022, Klecko was announced as one of the three finalists nominated by the senior committee for induction in the 2023 Pro Football Hall of Fame class.

[1] The young Klecko converted a '55 Chevy into a dragster and had a job driving dump trucks and oversized tractor-trailer rigs.

[16] In 2005, Klecko was involved in a fatal car accident, in which he accidentally struck and killed a pedestrian on the Major Deegan Expressway in Bronx, New York.