Hitmonlee can be obtained as a gift for beating the Karate King in the Fighting Dojo in Saffron City. You can obtain either Hitmonlee or Hitmonchan in this manner, but not both.
Ah yes, Hitmonlee, king of Hi-Jump Miss and many other such classics. Honestly, if you're even able to grab Hitmonlee without sacrificing Lapras thanks to your ruleset, it's probably not too worth it, as Fighting-types tend to falter pretty hard just about at the point you get it. Yes, they hit the hardest, but you can probably just put Brick Break on everything and be done with Fighting coverage. So what reason would you have to want Hitmonlee on your team? Well, let's see. Hitmonlee is the second strongest Fighting-type in Kanto, only beaten by a slight margin by Machamp. What sets him apart from Machamp is a little bit of Speed and pretty much double the Special Defense. Mark my words, Hitmonlee can tank almost any Psychic with that stat. If he's at full health and doesn't get critted, that is.
Gym #5 - Koga (Fuchsia City, Poison-type): You can't really hit Koga with much of anything when using Hitmonlee. Maybe Muk can drop to a Mega Kick, otherwise no sell. Koffings and Weezing have too much defense, in stark contrast to Hitmonlee who will NOT enjoy taking SelfDestructs to the face.
Rival (Silph Co.): Some of the strategy here revolves around your starter. If you picked Bulbasaur, your rival will have not only Pidgeot and Charizard lobbing Wing Attacks at you, which is something you should steer dead clear of with Hitmonlee, even if you put Rock Slide on it, there'll also be an Exeggcute with Confusion. It's an Exeggcute and it's Confusion though, and I'll do the math for you to save you the work. You'll be fine. Alakazam can be taken out with Mega Kick, as its only attacking move is Future Sight, and Gyarados has Intimidate so don't risk it. If you picked Charmander, the only things that can harm Hitmonlee in this fight are Pidgeot and shenanigans, since Exeggcute got Hypnosis instead of Confusion. Growlithe has Intimidate like Gyarados, so ehh... If you picked Squirtle, you have to deal with Pidgeot, two Intimidators and Venusaur. The deal with Venusaur is that it very much enjoys stalling and has decent defenses. And it resists your STAB. So no. Just take out Alakazam with a Mega Kick and be done with it.
Giovanni (Silph Co.): Any STAB turns Rhyhorn into a pile of rocks on the floor, and Kangaskhan is easily handled too, unless you miss. You're also probably safe taking out Nidorino with a Mega Kick. Nidoqueen is highly dangerous.
Gym #6 - Sabrina (Saffron City, Psychic-type): Remember how Hitmonlee can tank almost any Psychic? Well, this is one of the exceptions. Sabrina is very much one of the hardest battles in the game, Hitmonlee or no Hitmonlee. How can Hitmonlee help then? Two words: Mega Kick. You'll probably need to rev up with some X-Accuracy, maybe throw in an X Sp.Def for good measure, but once that's done, tough out Alakazam's Psychic and kick it in the face. It'll work if it doesn't outspeed and crit, and your health is still full. You can also probably handle Venomoth with Rock Slide.
Gym #7 - Blaine (Cinnabar Island, Fire-type): The dogs have Intimidate and the horses have Bounce. All-around not worth it.
Gym #8 - Giovanni (Viridian City, Ground-type): Rhyhorns are weak to your STAB and Dugtrio's Defense is pretty low (so is yours though, so watch out, especially since its Speed is insane). Leave the Nidos for something else, preferably someone who can handle Earthquakes.
Rival (Route 22, pre-Elite Four): Seeing how your rival's Pokémon tend to learn moves at whatever levels they please, Pidgeot now has Gust AND Wing attack. No-go again. Charizard still has Wing Attack too. Exeggcute can now only attack you with, and this is seriously serious, SolarBeam. You'll be fine. Gyarados and Growlithe have Intimidate still, and Alakazam is now extremely treacherous because you have little time to set up like you did on Sabrina. Venusaur is no longer a stall (well, in a sense) so you can take it out with Mega Kick, and Blastoise forgot all its good attacks at home, so it's a safe opponent too. You can probably fathom that Rhyhorn will obviously be the least of your worries.
Elite Four Lorelei (Indigo Plateau, Ice-type): Your STAB attack of choice will turn Dewgong and Lapras into Deadgong and Deadpras. Jynx's defense is bad enough to drop to Mega Kick or maybe Rock Slide if you nabbed it, you needn't worry about speed or Psychic thanks to your formidable Special Defense. Slowbro is a no-go though, thanks to that monstrous Defense. While Cloyster also has a monstrous Defense, it's way more of a wall/stall than a bulky offender like Slowbro is, so a wallbreaker like Hitmonlee should have little issue with it.
Elite Four Bruno (Indigo Plateau, Fighting-type): Both Onix have the Attack stat of a Spearow, and are weak to your STAB. Feel free to have a Hitmonlee mirror match, or perhaps have Hitmonlee take out his Hitmon-brother. Machamp isn't a safe enough bet though.
Elite Four Agatha (Indigo Plateau, Poison-type): Nothing Agatha has can be reliably defeated by Hitmonlee. Not a single one of them. Ghosts? Haha, no. Golbat? Haha, no. Arbok? Intimidate. All-around just no.
Elite Four Lance (Indigo Plateau, Dragon-type): Only the Dragonairs can really be taken care of by Hitmonlee without much issue. Even with Rock Slide, you shouldn't be taking on the others. Gyarados has Intimidate, pretty much like every other Gyarados in the game, and the others can probably outspeed you and smack you in the face with Wing Attack.
Champion Rival (Indigo Plateau): I'll give you a run-down here because there's not a lot of strategy to discuss. If you've been paying attention earlier, you should know how to deal with most of this battle already. Pidgeot - no. Alakazam - maybe. Rhydon - easy. Exeggutor - Mega Kick. Gyarados - Intimidate. Arcanine - Intimidate. Charizard - no. Blastoise - watch out for Hydro Pump. Venusaur - slightly dangerous.
Post-Game: Mewtwo? HAHAHA. The League rematches follow essentially the same rules as before, instead.
The big decision is which Fighting-type STAB you want: Brick Break or Hi Jump Kick (comes a level after you get Hitmonlee, at level 26). Hi Jump Kick can be described as Hi-Risk, Hi-Reward, while standard Brick Break is a bit weaker, but a hell of a lot safer and more reliable with its 100% accuracy. You'll also want to grab Mega Kick (level 46 or Route 4 Move Tutor), which is inaccurate but powerful as hell. You also get the choice between Rock Slide and Earthquake, both of which are probably better off on something else, but hey, if you want to have a Hitmonlee, you might as well roll with it. The last move comes down to a matter of preference. Do you want Bulk Up for the Defense boost, or do you want Mind Reader (level 31) to ensure your killing power? You could also grab Strength for some HM-age and more assured hits when Mega Kick isn't safe enough. Hitmonlee can also learn the Substitute and Focus Punch combo for an insanely powerful STAB move that really isn’t worth it unless you’re in a boss battle. Return can work over Mega Kick, as it has better accuracy and PP but less power.
Recommended moveset: Hi Jump Kick / Brick Break, Mega Kick / Double Edge / Return, and any two between Rock Slide / Bulk Up / Brick Break / Earthquake
Minimum stats are calculated with 0 EVs, IVs of 0, and a hindering nature, if applicable.
Maximum stats are calculated with 252 EVs, IVs of 31, and a helpful nature, if applicable.
What Nature do I want? Most of them are good. Anything that lowers Attack or boosts Special Attack (or possibly both, but screw Modest anyway) is bad. Adamant and Jolly are probably the best. Careful is also good, as it improves your special tankiness and sacrifices virtually nothing. Speed isn't too important, so you could afford to sacrifice some of it with Sassy or somesuch.
How good is Hitmonlee in a Nuzlocke? Not too great. You probably have better options. If you put Hitmonlee on your team, there's really only three reasons. Either you're doing it out of the kindness of your heart, you want to bring some variety to the forums, or it's the best backup in your box. Stat-wise, Hitmonlee is pretty survivable, and has some nice stopping power offense-wise.