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Go to the house north of Cianwood City. Go to Route 42, outside Mt. Go to Vermillion City. Go to Route 14 of Kanto. Go to Cerulean City. Battle Suicune. Throw Dusk Balls and/or Ultra Balls until Suicune is captured.
Suicune is to the right of the house at the north of this island but will vanish when you get close. Suicune isn't like Raikou or Entei since it can't be caught randomly. You have to go to certain locations to encounter Suicune before you can capture it. You also don't need to prevent Suicune from running from battle, so you won't need your Snorlax or Gengar. Mortar. You should be able to surf west a bit and find a tree you can cut. It will lead you to a small grove with three trees with berries in the middle. Suicune will be there, but will again run away when you get close. Suicune will be there on the port leading off of the S.S. Aqua, but will disappear when you get close. It's towards Route 13. You'll find Suicune next to Bill's house on Route 25 after you've defeated Misty and she returns to her gym. The Legendary Pokemon is level 40 and has the moves Rain Dance, Gust, Aurora Beam and Mist, so make sure you have a lead Pokemon that can take his hits. Since Suicune is a Water-type Pokemon, you'll want to use a Grass, Water, or Dragon-type Pokemon. Use False Swipe to keep Suicune at 1 HP, so you don't accidentally defeat it. If you defeat Suicune, that's ok, go heal your Pokemon at the nearest Poke Center and return to the burned tower.