Endowed with great natural strength, quick feet and an enormous desire to win, he anchored the Jets formidable defensive line known as the "New York Sack Exchange", along with Marty Lyons, Abdul Salaam and Mark Gastineau at left end. The four combined for 53.5 sacks in 1981, and Klecko is second to Gastineau on the Jets all-time leading sack leaders (as of the 2005-2006 season). His son, Dan, plays for the New England Patriots.

While at Temple University, Klecko also played semi-pro football for the Aston (Pa.) Knights and kept his eligibility by playing under an assumed name (Jim Jones) and a fake university (Poland University). He also won two NCAA club boxing titles in the heavyweight division.

On Sunday, December 26, 2004, in a halftime ceremony, the New York Jets have honored the first player in professional football history to be selected to the Pro Bowl at three different positions (DE, NT and DT) by retiring his #73 jersey. Klecko became just the third New York Jet to have his number retired, joining Hall of Fame QB Joe Namath and Hall of Fame WR Don Maynard.

Hall of Fame centre Dwight Stephenson, a player with the Dolphins who faced the Jets twice each season for many years, considered Klecko and Hall of Famer Howie Long his two most difficult opponents. But for an injury-shortened career, Klecko might already have been selected for the Hall and has appeared as a finalist in recent ballots.

On Monday, September 13, 2005, Klecko accidentally hit and killed Ray Peterson, a 65-year-old man pushing a grocery cart, with his car while on the Major Deegan Expressway. No charges were brought.

Klecko had a small role in the 1981 Burt Reynolds film Cannonball Run, as well as Reynolds's Smokey and the Bandit II.