Pretty much the entire "Wings of a Dope" episode, in which Buckley's angel comes back, particularly the scene in which he and Luanne bounce on the trampoline after first meeting. Fireflies appear around them and fittingly enough, Dream Academy's "Life in a Northern Town" plays in the background. The last scene is even more powerful - Buckley bounces on Luanne's trampoline high enough to disappear from her sight, after which she believes he is gone for good and goes back into the house. He returns to Earth, exclaims, "Cool, a new record!" and then, as the song begins to play again, is shown walking off toward the horizon, pulling a halo from his pocket and donning it as the credits begin to roll.
The scene where Luanne is by the trampoline crying listening to "Angel Flying Too Close to the Ground" as she waits for Buckley and hopes he isn't "gurardianin' some other girl," followed by Peggy's motivational speech, is a close second.
Meta example: "Wings of the Dope" premiered two weeks to the day after the Columbine shooting. Mike Judge received a letter from a young woman from Littleton about a month after the episode aired. She went to Columbine, had a crush on a classmate she knew, and promised herself during the shooting that if both survived, she would tell him how she felt. She never got the chance - he was one of the shooters. Like Buckley's final birthday card to Luanne, "Wings of the Dope" was what finally prompted the girl to open up and grieve after pressure to bottle up her feelings from people who believed the boy didn't deserve it. To turn the tearjerker factor Up to Eleven, someone who knows the girl has said that the girl quoted Luanne in the letter and hoped he wasn't "guardianin' some other girl", since he never knew how she felt.
"John Vitti Presents: Return To La Grunta" for its handling of sexual harassment. Its realistic portrayal of the stigma against teporting, especially for men, hits home to many survivors of sexual assault and harassment. Even better is Hank's Crowning Moment of Awesome when he throws Luanne's harasser into the pool and then stimulates the dolphin, causing it to attack the offender.
Almost any episode from the first few seasons which focuses on Luanne. Whether it's the reveal of her mother's alcohol problem, general discussion of her dysfunctional family, her futile struggles with beauty school, her boyfriend's death and return as an angel, or her facing sexual harassment at work, the early Luanne episodes tended to be pretty heavy.
There's also the episode "Propane Boom," the finale of the second season. When the Mega Lo Mart explodes as a result of a negligent propane leak, Peggy screams in terror for Hank and Luanne, who're both inside. The scene is not Played for Laughs, with the season ending upon Boomhauer calling 911 and a horrified Peggy watching him do so, unaware if her husband or pseudo-daughter are alive.
The following episode has Luanne dealing with the loss of her hair and bottling her emotions by pulling a Sinead O'Connor act, but she soon breaks down alone in her room after reading the last birthday card Buckley gave to her.
Hank having to deal with possibly euthanizing Ladybird in "To Kill a Ladybird". He closes the garage door and turns on his power tools to drown out the sound of him crying, though that doesn't quite do the job.
Peggy lamenting about how each of her birthdays always ends in disaster in "Strangeness on a Train".
Peggy crying when she finds out that her big feet were being used to satisfy Internet perverts on "Transnational Amusements Presents: Peggy's Magic Sex Feet."
Bobby telling Peggy that she shouldn't feel bad about her big feet just because the perverted podiatrist said so and the fact that Bobby accepts the fact that he's fat (or "husky" as his jeans says) and there are people out there who pick on him for it, but he doesn't let it get him down because he has friends, a girlfriend (at least until he and Connie broke up), and most people see past the fact that he's "the fat kid" and see that he's a very funny, friendly person.
The ending of "Won't You Pimai Neighbor" where Bobby chooses what's in the mirror- Connie's reflection- meaning he doesn't want the celibate life of a monk if it means he can't be with her. And it turns out the mirror was the object belonging to the Lama, but the monks decided that the choice was still legitimate.
Connie's reaction, shedding a single tear of happiness, was what got to me. While divisive for the King of the Hill fanbase in general, Bobby and Connie's breakup in a later episode.
Quite a few Ladybird-centric episodes have their moments, but "To Kill a Ladybird" takes the cake. Ladybird is feared to have rabies and Hank, while grappling with the fact that he may have to put her down shuts the garage down and turns on some loud power tools which fail to mask Hank audibly sobbing.
Look at what you do! You made me think about that, we all know how Hank is...and him crying about Ladybird is the only time anything on TV has ever made me cry.
"Manger Baby Einstein" had the Gurgle-Gurgle subplot.
"Pretty Pretty Dresses". Bill is already a horribly depressed and thus, to the audience, depressing character, but watching him break as thoroughly as he does here is really awful. At least it leads to a Crowning Momentof Heartwarming.
The series finale is too much to watch, I watch the show all the time, and it always feel like it's still going on and will never end. The realization at the end of the finale makes me break down every time. This really is one of the greatest animated series ever made. If you've seen every episode, you get attached to the characters. Just thinking that we'll never see them again really hurts.
When Peggy is about to break to Dale about how Nancy has been cheating on him all these years with John Redcorn, when Joseph (John Redcorn's biological son) comes out of the house and asks Dale for some money to see a movie. Dale gives him extra for snacks, and you realize how much Dale loves Joseph (even though Joseph is biologically not his). Peggy can see this too and decides not to tell the truth.