Instructions
Complete the game once through and collect an electric Pokemon. Also make sure you have the ability to surf, which you should collect over the course of the game, and purchase at least ten Ultra Balls from the various in-game shops, which are extremely powerful Pokeballs.
Go to Fuschia City and head far west toward the sea shore, where you'll find two people chatting.
Use your Surf ability to swim over the water and head southwest until you hit the Sea Foam islands, where you'll find a cave.
Enter the cave, head to the second floor and push the two boulders you'll find into the little holes in the cave floor. This will stop the water currents, allowing you to swim through the previously inaccessible stream of water on the first floor.
Swim through the now-gentle water until you reach a little piece of land; step onto the land to battle Articuno.
Fight Articuno using one of your electric Pokemon to counteract its water/ice power until it is very weak, and then throw Ultra Balls at it until you have captured it.