Instructions
Beat the Elite Four and the Pokemon Champion. Latios will only appear after you accomplish this task. Look for Latios in the long grass of any route. Eventually, it will appear. Once it does, it will flee within a few turns.
Fly to any city with a PokeMart. Purchase 40 to 50 Ultra Balls, which you will use to capture Latios. Teach a Pokemon Bug-, Dark-, Dragon-, Ghost- or Ice-type moves, since Latios is susceptible to these. Get a Pokemon that has the "Shadow Tag" ability or knows "Mean Look." Both Pokemon should be around level 40. Put the Pokemon that has the ability "Shadow Tag" or the move "Mean Look" as the first one in your party.
Fly to any city. Go to a nearby route and return to the city. Check your Pokedex. Each time you change locations, Latios will go to a different route.
Leave the city, return and check your Pokedex until Latios is in the same location as you. Once it is, save your game. Look for Latios in the long grass. Use "Mean Look" on Latios as soon as you see it. If you have a Pokemon with the "Shadow Tag" ability, you won't have to do anything to prevent Latios from running away.
Harm the Latios with Bug-, Dark-, Dragon-, Ghost- or Ice-type moves. Use Ultra Balls once it has less than 20 hit points left. Induce a status effect, such as paralysis, by using moves like Spark. A status effect will make Latios easier to catch.