Instructions
Go to Sootapolis City. Either dive from Mossdeep Town and travel through route 229 until you find the cave that goes there, or fly over there with a Pokemon that knows the Flight skill.
Dive around the waters of Sootapolis City. Relicanth is a water-type Pokemon, and therefore can only be found underwater.
Journey around the grass areas underwater around Sootapolis City. Pokemon only appear in grass patches, and Relicanth is no exception. You might have to do this for awhile, because there is only a 2% chance of him appearing.
Weaken Relincanth as much as possible when you begin fighting him. Casting sleep or any negative status ailments will also help in catching him.
Throw any Pokeballs that will increase your chances of a capture. Ultraballs are standard issue, but the better ones are Diveballs, which are specifically made for a water-type Pokemon such as the Relicanth. If you don't want a challenge, throw the Masterball to catch it.