Gideon Lyon is like any other criminal who goes about doing crime waves and whatnot. He's also known for being manipulative and a stalker.
He took his job as a CIA agent way too seriously than most agents would serious enough. Somewhere along the line, he had some bad reputation held against him, which led him to being fired from the agency. That was when he took the job as the right hand man of Tito Flores and operative for the cartel.
Gideon Lyon used to be in the CIA, and then he had gone sour and ended up working for the cartel of Tito Flores as an operative.
It's been revealed that Gideon is the one responsible for the death of Miranda, Riggs' late wife. Right when Riggs received note of it, he went after Lyon and Flores in Mexico.
Catching up with Lyon, Riggs captured him and interrogates him, torturing him in the process. Lyon admitted he was responsible for Miranda's death. After her father, Ronnie Delgado, refused to cut a deal with Tito Flores in letting them off the drug charges, Flores sent Lyon to go after Miranda for revenge in this. He also admitted to Riggs that he had been spying and stalking on them at their house back in El Paso and was waiting for the right time to wreck Miranda in her car while she was on her way to the hospital. She didn't die right away but was barely able to breath, which left Gideon enough time to suffocate Miranda and then covered his trails to look like it was a car accident. After being tortured further by Riggs, he gets stuffed into a safe place by Riggs so he can deal with him later. Sometime after, Lyon got loose and waited for the opportunity to take down both Riggs and Murtaugh.
After capturing them both and taking them to a hideout near a subway station, he tortures them one by one. Riggs somehow manages to get free from his bonds, and throws the knife at one of Lyon's henchmen, and takes one of their guns and shoots one by one, with Lyon escaping. Riggs goes to catch up with Lyon somewhere in the station, and Lyon comes out attacking him. There, they get into a struggle, and they both fall onto the subway train tracks, where Riggs gaining the upper hand on Lyon, just moments before the oncoming train was coming, which Murtaugh was warning Riggs of. At the last moment, Riggs got off the tracks with the train running on Lyon.