Zapdos Can be found at the Kanto Power Plant at level 50. If you intend to use it, make sure to bring several Repels, the Master Ball, or a whole lot of Ultra Balls and and some serious guts.
Zapdos holds the fortunate position of being one of the earliest obtainable legendary Pokémon for use in game, along with having an excellent typing of Electric/Flying, making it an excellent choice for many players. This is no different in a nuzlocke; the only flaw of Zapdos in a nuzlocke is the difficulty not of finding it, but catching it, particularly without any losses.
Gym #3 - Lt. Surge (Vermilion City, Electric-type): If you've put off Surge for this long, either you completely forgot about him, or you really, REALLY hate his guts. Zapdos can easily destroy Surge thanks to an insane level advantage, but you should already be incredibly overleveled for this fight if you have Zapdos.
Gym #4 - Erika (Celadon City, Grass-type): Same situation as Surge, really, except now you also have a type advantage thanks to STAB super effective Drill Peck. Again, though, you should be really overleveled anyway if you have Zapdos for this fight.
Fighting Dojo (Saffron City, Fighting-type): Finally, a fight where it makes sense to have Zapdos available for it. Just Drill Peck everything, though you may want to just let someone else deal with this place for the experience.
Rival (Silph Co.): Use your Electric STAB on the Pidgeot, resident Fire-type, and resident Water-type, Drill Peck everything else.
Giovanni (Silph Co., Ground-type): You should really just avoid the Onix and Rhyhorn, but everything else will be easy to handle with a few Drill Pecks. You could teach Zapdos Steel Wing if you want to deal with the Rock-types, but it's not worth it; a strong enough Hidden Power Grass or Water would probably do better, if you have that.
Gym #6 - Sabrina (Saffron City, Psychic-type): Spam Drill Peck. You'll outspeed even the Alakazam thanks to the level advantage, and their low defenses won't let them take a STAB Drill Peck worth beans.
Gym #7 - Blaine (Cinnabar Island, Fire-type): This fight is a pretty neutral matchup, and you no longer have that level advantage to ride off of. If you have Thunderbolt, you can handle this (Intimidate from Arcanine and Growlithe plus potential to be burned make Drill Peck less helpful), but you need a surfer not only to reach Blaine, but also Zapdos, so you have almost no excuse for not just using that.
Gym #8 - Giovanni (Viridian City, Ground-type): Same situation as the last fight with him, really. Spam Drill Peck against anything that isn't Rock and let something else handle the rest.
Rival (Route 22, pre-Elite Four): Same as last time, except you should avoid the new addition, Rhyhorn.
Elite Four Lorelei (Indigo Plateau, Ice-type): Spam Thunderbolt on everything, but Lapras might survive and hit you with Confuse Ray or Ice Beam, so don't be reckless. Potential Rain Dance sets are also better off not using that move in this case, for obvious reasons.
Elite Four Bruno (Indigo Plateau, Fighting-type): Avoid the two Onix, Drill Peck everything else.
Elite Four Agatha (Indigo Plateau, Poison-type): Thunderbolt spam will work quite well, but STAB Sludge Bomb from Gengar will hurt a good deal more than you'd think. Bring healing items in case you need them.
Elite Four Lance (Indigo Plateau, Dragon-type): Gyarados will go down to a single Thunderbolt. Aerodactyl is a risky matchup, as it has AncientPower and outspeeds, but it will go down in one hit if you use an Electric attack. The two Dragonair can be handled with Drill Peck, but Thunder Wave will not be fun as Zapdos isn't immune to paralysis in this generation. The Dragonite is best left to something with Ice Beam, as STAB Outrage really hurts.
Champion Rival (Indigo Plateau): Honestly, the same deal as the other fights. Spam Thunderbolt or Thunder against everything that doesn't resist it, use Drill Peck for the rest unless it's Rhydon, and leave Rhydon to something more suitable.
Post-Game: Maybe later. At any rate, Zapdos is not bad against Mewtwo.
When you first catch your bird of thunder, it will know Thunder Wave, Agility, Detect, and Drill Peck. Other than these moves, it learns little by level up. It learns Charge at level 61, which is completely useless in Generation 3; Light Screen at level 73, which is nice, but far too late and available via TM anyway; and Thunder at level 85, also far too late and available by TM. The Move Reminder will give it Peck and ThunderShock, but there is no reason to learn these.
Through TMs and HMs, things get much better, giving it a still small but good movepool. It can, of course, learn Thunderbolt, but if you would prefer not to grind for the money to get that TM, Thunder can actually work quite well, as Zapdos has the bulk to survive a hit if it misses. You could also pair this with Rain Dance for better accuracy, especially if you have a few other Pokémon that can benefit from rain. It also learns Sandstorm and Sunny Day, but it has no use for those moves itself. Steel Wing, Hyper Beam, Facade, Hidden Power, Secret Power, and Return can all provide some extra coverage, but Return sees more use on something that receives STAB from it or has better Attack than Special Attack. Zapdos learns Roar, Toxic, Protect, Double Team, Rest, and Light Screen for a supportive role, but you're usually better off attacking. Finally, it can learn Fly, Flash, and Rock Smash for utility if you need something to learn them, as Electric/Flying STAB can cover enough anyway.
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.
Zapdos Line Ratings
What Nature do I want? Special Attack boosting natures are best, but Attack lowering ones cut the power of Drill Peck. The nature doesn't matter much one way or the other, though; in the end, Zapdos makes a good use of all of its stats.
How good is Zapdos in a Nuzlocke? If you catch it, very, VERY good. Guaranteed encounter, great bulk, speed, and power, and very nice typing makes it a real terror.
Weaknesses: Ice, Rock
Resistances: Grass, Fighting, Flying, Bug, Steel
Neutralities: Normal, Fire, Water, Electric, Poison, Psychic, Ghost, Dragon, Dark