Instructions
Spot the Suicune in the north side of Cianwood City. It will run away.
Spot the Suicune in Route 42. It will be just outside Mount Mortar, and it will run away again.
Spot the Suicune in Vermillion City in Johto. You will see it as you get off the boat to Vermillion City, and it will run away again.
Spot the Suicune in Route 14. It will run away for the final time.
Go to Route 25 and find the Suicune one last time. You will now be able to battle and capture the Suicune. Use Electric or Grass-type Pokémon against it, as well as Electric and Grass-type attacks. Make sure the Pokémon you use to attack it is between levels 35 through 40. For the best results, use an Ultra Ball to capture it. You may have to throw the Ultra Ball at it several times to successfully capture it. Do not knock the Suicune unconscious.
Spot the Suicune in Ecrtueak City at the Burnt Tower if you accidentally knocked the Suicune out in Route 25. Battle and capture the Suicune. One effective method is to use the Sleep Powder attack on it and then use an Ultra Ball to capture it.