He appeared in the episode "Death and Texas".
Wesley Martin Archer is a prisoner on death row for killing a man after breaking in by shooting him and blowing his head off. During his time in prison, he would write to every teacher in the Arlen yearbook telling them that "they had a positive impact in his life and to teach him". Peggy Hill responded to the letter and came to visit him in prison, where he tries to make himself an innocent man who accidentally killed the man and wants to learn how to read. Peggy agrees to do so. During the visits, Wesley would win her over by drawing an apple for her and pretending to attack a guard that he claimed was sexually harrassing her, an act that would get him three days in the hole. Eventually she helps him by teaching him how to play Boggle. In the off-time, the timer gets broken after a fight and claims he needs more timer sand. Peggy would pay a visit to his friend Janet and gives her a bigger timer for him to use, claiming he's slow and needs more time. She gives the timer for Wesley and in the off-time gets broken and wants more timer sand. Peggy goes back to Janet and gets a brick of sand for him. Later on, when alone, Wesley confesses to Peggy that it's not really timer sand but it's cocaine, which smuggling it to him was a federal offense and also said that Peggy never teached him and he grew up in Arkansas and was pushing 40. He then says he wants cocaine every week alongside other demands. Peggy stops showing up to the prison. Wesley calls the Hill residence and says if he doesn't get his cocaine, he's going to turn in Peggy for fun. Peggy refuses, and dreads her going to prison for being used as a drug mule. He turns her into the Warden, who then calls Peggy into the prison and wants a confession. Peggy notices the timer was empty and licked clean (Wesley also tasted the cocaine to make sure it's pure before giving it to the Aryan Brotherhood). Peggy uses this to her advantage and lies and says she never smuggled drugs into the prison. Angered that his plan is foiled, he attempts to attack Peggy, but due to him being shackled, gets taken away by the guards.
It can be assumed later on he is executed.
The character name comes from Wesley Archer, who served as supervising director on several King of the Hill episodes as well as The Simpsons.