When you wake up the Legendary Dogs at Ecruteak City, the Legendary Dogs will escape in to the wild. The question you probably had in mind is: where? Well, they can be anywhere. The Legendary Dogs appear completely random in the entire map of Johto. Don't worry about Kanto because they don't appear there. The Legendary Dogs can pop out anywhere in the wild in the entire map of Johto, and there's no telling where they can be. However, once you do see one of them. That one you saw will be entered in your Pokedex. Using your Area feature on your Pokedex, you will be able to see where the Pokemon is. But the Legendary Dogs keep moving constantly. If you don't get to where they're quickly, chances are that they will be somewhere else. Believe it or not, that's actually the easy part. Yeah, the easy part. Capturing them is the hard part. When you do encounter one of the Legendary Dogs, it will frequently run away immediately. Making capturing very difficult. But if you did damage to the Pokemon, the HP bar will still be the same the last time you saw it. Be careful to not knock it out because once they're are knocked out, they will not appear anymore. Don't save your game and turn it back on if you need to.
If you haven't seen any of the Legendary Dogs yet, then its going to take some time until you encounter it sometime. I was training my Pokemon in the grassy areas when I first encounter one of the dogs. When i used my Pokedex to track it down, I luckily bumped into a different Legendary Dog. If you have seen at least one of the Legendary Dogs, then I can offer you some tips. Ofcourse, you will be using your Pokedex's Area feature to track down where the Pokemon is. Make sure you have a Pokemon that knows Fly. Pick a random city to fly to because once you fly the Legendary Dog will change locations, in most cases. So it doesn't matter if you know where the Pokemon is, once you use Fly, it will be at a different place, most of the time. Keep repeating this process until the Pokemon is somewhere near you. If the Pokemon is near, quickly rush to the place where the Pokemon is. Keep searching and run away from other wild Pokemon. Check your Pokedex once in a while to see if the Legendary Dog is still there. If the location is changed, do the same thing over again. Fly to some city, and check Legendary Dog's location. And that's just finding it. When you do all this, you should have a Pokemon that causes sleep. Put that Pokemon in front of your lineup so that you can immediately induce sleep on the dog before it quickly runs away. This Pokemon you use should be at a high level so that it will attack first and withstand attacks. Of course, knock the Pokemon's HP down as low as possible. Remember, the Legendary Dog will keep its HP from the last battle it was in. So if the Legendary Dog runs away, any damage done to it stays. If you want to use a Master Ball you don't need to go through this, you just need to throw the ball. Otherwise, Ultra Ball is your best bet.
Alright, the Legendary Dog's HP is too low to attack once more, but its still difficult to capture. Well, your best chances on capturing a Pokemon is if the Pokemon is asleep. Sleep is a must. Paralysis works good, but not as good as sleep. This is also very important: carry a low level Pokemon with you so that you can do very minimal damage if needed. If the Legendary Dog has little HP left, but can still go a bit lower, you don't want to use your Lugia's Aeroblast. So having a low level Pokemon with you with weak attacks can do some small damage. You have to make the Legendary Dog asleep first, or it will run away or your Pokemon will be toast. It all comes down to your skills, good luck!