Raikou/Suicune/Entei Factor
After you defeat the Elite Four and have your Pokédex upgraded to National Mode, one of the three legendary dogs Raikou, Suicune, or Entei will roam around Kanto; not The Sevii Islands. The legendary dog that appears depends on which Pokémon starter you choose at the beginning of the game:
Factors
If you choose Bulbasaur, Entei will roam around Kanto
If you choose Charmander, Suicune will roam around Kanto
If you choose Squirtle, Raikou will roam around Kanto
Capturing
If you remember from Gold/Silver/Crystal, these Pokémon would be found randomly around Johto. The same is true in Kanto, and yes; they'll still run away the first chance they get. You'll know when you've found either Raikou/Suicune/Entei when you hear different battle music than normal. Just finding Raikou/Suicune/Entei the first time is the hard part. Afterwards, you can use the Pokédex "Area" tool is locate the legendary dog. Bear in mind though that whenever you move to a different location, it will also move. It can take a little effort for Raikou/Suicune/Entei to show up on the same route as you.
If you have the Master Ball still, this would probably be the best time to use it. Otherwise, in order to keep the legendary dog from running you'll need to use a move like Mean Look to keep it in battle. That, and make sure the Pokémon using the move has a high speed stat, or is holding the quick claw to prevent Raikou/Suicune/Entei from just running away right from the start. If you inflict damage to Raikou/Suicune/Entei and then it flees, that damage will remain; they can't heal themselves. Don't KO Raikou/Suicune/Entei by mistake, for there's only one.
After you've kept it in the battle, just whittle it's HP down, and have many many Ultra Balls to capture it. With a little time and patience, you will.
