Falkner's only anime appearance was in the episode Fighting Flyer with Fire.
Falkner is obsessed with Flying Pokémon and spends most of his time either gliding around on a hang glider or perched with his Hoothoot on his shoulder. When Team Rocket steals Ash's Pikachu early on, he swoops in to save the day with his Pidgeot—which pretty much sets the tone for how he operates.
He runs the Violet City Gym, a towering spiral building where all his battles take place on the roof. His matches are a whole production; he's got a crew of students who chant in unison like it's a stadium event. And yeah, he's got that self-righteous vibe about Pokémon training that gets under Ash's skin immediately. He's particularly passionate about defending Flying-types against their Electric weakness—though he doesn't hesitate to call out Ash for bringing a Chikorita (weak to Flying moves) into their match, which comes across as pretty hypocritical.
Despite his Charizard taking serious wing damage during the battle, Ash managed to pull off the win and claim the Zephyr Badge.
Content compiled by AnimeList.moe from publicly available sources.