Articuno can be found as a static encounter in the Seafoam Islands at level 50. If you want to catch it, you'll need a lot of Repels, and preferably the Master Ball. If you don't have the latter, then bring a lot of Ultra Balls, and some healing items to keep your Pokémon alive while trying to catch this legendary Pokémon.
Articuno is a member of the legendary bird trio. Just like every other legendary Pokémon, it has quite high stats, comes at a high level, and is only available late in the game. However, it is really worth it, as it has an advantage against nearly everything in the League. The most difficult is to actually catch Articuno. This won't be a problem with the Master Ball, but if you don't have that anymore, be careful, as Ice Beam hits hard. Bring a bulky Water-type if you truly want to catch it, and lots of healing items too.
Gym #3 - Lt. Surge (Vermilion City, Electric-type): Okay, you really shouldn't have delayed this for so long. Ice Beam makes mincemeat of all of Surge's Pokémon, thanks to being massively overleveled.
Fighting Dojo (Saffron City, Fighting-type): Teach Articuno Fly, and sweep through the gym. Long live being overleveled.
Rival (Silph Co.): Ice Beam beats Venusaur, Exeggcute, Alakazam, Gyarados and Pidgeot, Fly obliterates Growlithe and Blastoise. Charizard can use Flamethrower, but you have a level advantage of roughly, what, ten levels or even more. Ice Beam takes it out.
Giovanni (Silph Co., Ground-type): Haha. Ice Beam everything for victory.
Gym #6 - Sabrina (Saffron City, Psychic-type): Ice Beam everything again. You could use Fly against Venomoth too, but honestly, it's not like anything would be able to harm you with this level advantage.
Gym #7 - Blaine (Cinnabar Island, Fire-type): All of Blaine's Pokémon know Fire Blast...but sadly, Growlithe and Ponyta are incredibly weak. Articuno can use Fly or Water Pulse to take them out easily. Rapidash and Arcanine should be handled by something else.
Gym #8 - Giovanni (Viridian City, Ground-type): Heheh, Ground-types. Ice Beam everything again, and take the victory home.
Rival (Route 22, pre-Elite Four): Ice Beam rips through Venusaur, Exeggcute, Pidgeot, Rhyhorn, Gyarados and Alakazam. Fly takes Growlithe and Blastoise out. Don't bother battling Charizard, that Flamethrower will hit harder than you want it to.
Elite Four Lorelei (Indigo Plateau, Ice-type): Fly can beat Dewgong, but avoid battling Cloyster and Slowbro. Both have insane Defense stats, and the former can use Spikes to hurt your incoming Pokémon as well. Jynx can be beaten if you taught Articuno Steel Wing. If not, then Fly should take it out as well. Lapras is too bulky, and hits hard with Ice Beam and Body Slam. However, if you have healing items, you should be able to sweep this fight, as none of Articuno's harder matchups (Cloyster, Slowbro, Lapras) are able to knock Articuno out quickly, even with critical hits.
Elite Four Bruno (Indigo Plateau, Fighting-type): Both Onix get OHKO'd by Ice Beam, Hitmonchan and Hitmonlee by Fly. Watch out for Hitmonchan's Sky Uppercut while you're in the air. You might want to skip Machamp, as it knows Rock Tomb, and Articuno has a 4x weakness to Rock-type moves.
Elite Four Agatha (Indigo Plateau, Poison-type): Articuno honestly doesn't care about any of Agatha's Ghost-types. Ice Beam or Fly sends them into oblivion. Just heal when needed, as they focus on using status moves. Ice Beam Arbok as well. Even a super effective critical Iron Tail from Arbok wouldn't be enough to take out Articuno, and Ice Beam 2HKOs it. Golbat gets beaten by one Ice Beam.
Elite Four Lance (Indigo Plateau, Dragon-type): Gyarados can't hit Articuno hard, and gets beaten down by Ice Beam. As for Lance's two Dragonair and his Dragonite, Ice Beam OHKOs them. Avoid Aerodactyl, as it will probably outspeed Articuno, and hit hard with AncientPower.
Champion Rival (Indigo Plateau): Start off by using Ice Beam to destroy Pidgeot, Rhydon, Exeggcutor, Gyarados, Alakazam and Venusaur. Fly will eventually take out Blastoise. Don't battle Arcanine or Charizard, as they boast Flamethrower and Fire Blast, respectively. Use healing items if necessary.
Post-Game: Articuno keeps at it. Mewtwo gets beaten by it, at least. Just don't send it out against anything that knows Fire-, Rock- or Electric-type moves that it can't OHKO.
Articuno at level 50 knows Mist, Agility, Mind Reader and Ice Beam, of which Ice Beam is the only really notable move. Reflect is learned at level 61, which is a good supportive move. At level 73, Blizzard comes, but this will be too late for the first League challenge. The last move will definitely come too late. It is Sheer Cold, which only comes at level 85. This gives Articuno the possibility to abuse the infamous Sheer Cold/Mind Reader combo, but it's not like you won't be able to OHKO almost everything at that point anyway, though.
All of Articuno's useful TMs can already be obtained by the time you caught it. Water Pulse comes from Misty, but is overall not too useful. Toxic is obtained from Koga. Articuno has the bulk to abuse it, but mostly its attacking stats are powerful enough to go without Toxic. Ice Beam can be obtained in the Celadon Game Corner, if you forced Articuno to forget it for some reason. Blizzard can be obtained in Pokémon Mansion, if you need a more powerful alternative to Ice Beam. Usually, it isn't necessary, though. Reflect can be received from the girl on the Celadon Department Store rooftop, after giving her a Lemonade. Reflect is useful, as it boosts Articuno's Defense stat, which is 'only' 100. Aerial Ace is obtained on Route 4, and might be useful if Agatha's Gengar decides to use Double Team. Honestly, it has no usage besides that. Steel Wing can be found in the Safari Zone, but it only useful against Lorelei's Jynx. Lastly, there's Fly, the best STAB Flying-type move for Articuno.
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.
Which Nature should I be looking for? Anything works on Articuno. Fly goes off Attack, Ice Beam off Special Attack. It has both great physical and special bulk. And its Speed isn't stellar, but not bad either. Any Nature could work.
How good is Articuno in a Nuzlocke? Very good, just like a legendary Pokémon should be. It has the bulk to shrug off any hit that isn't super effective, whether it's a critical hit or not, and can dish them back out. Of course, its movepool isn't that great, and it isn't the easiest Pokémon to catch. But if you use Repels, it's a guaranteed first encounter.