Instructions
Proceed through the game until you've beaten the Elite Four. Once you're the champion, the credit will roll. After that, the game starts again. You're back in your bedroom.
Walk downstairs and use the flickering television. It'll give a report of a new species of Pokemon that's roaming the countryside. Your mom will ask what color the Pokemon is, and you can select either "Blue" or "Red." Select "Blue" for a Latios. "Red" results in a Latias.
Open your inventory and select the Pokedex. On the map of Hoenn, you'll see a blinking dot. That's the current location of Latios. He roams from route to route.
Travel to Latios's current location and then wander around in tall grass. You'll randomly encounter Pokemon, and if you're lucky, you'll fight the Latios.
Attack the Latios until its health meter turns yellow or red. It's weak against Dragon, Ice and Dark type Pokemon, so use them to deliver the attacks. Once its health is down, throw pokeballs at the Latios until you've caught it.