Instructions
Buy as many Repel items as you can from a Pokemart in the game. These can be used from the menu screen to keep wild Pokemon except Raikou away. This will increase your chances of running into him.
Enter the Burned Tower level in Ecruteak City. Make your way to the top, battling trainers and random Pokemon. At the top, fight your rival. This battle is different depending on your starting Pokemon. When he is defeated, enter the next room and approach the three statues. They will come to life and fly away. Exit the Burned Tower level.
Make your way to Cherrygrove City in the Johto Region, either by foot or the Pokemon "Fly" ability.
Exit to the left of Cherrygrove city and use a Repel item by selecting it in your item menu. Walk back and forth across the tall grass area nearest the city. When your Repel wears off, use another. Continue until Raikou is encountered. Raikou is actually marked on the map in your menu screen. Select the map and look for a blinking dot to get his exact location.
Weaken Raikou in battle without killing it. Use a "Sleep" causing move to put it to sleep. Use a strong "Pokeball" item, such as "Heavy Ball" or "Super Ball" to attempt to catch it.
Check your map if Raikou runs from battle. He will probably relocate to another area in the Johto Region. Fly to the nearest city and walk through the grass until he is encountered again.