Buy lots of Dive Balls (at least 25) at the PokéMart in Mossdeep City. These balls are 3 1/2 times as powerful as a regular Poké Ball against underwater Pokémon (e.g. Relicanth); that's almost twice as powerful as an Ultra Ball.
Bring a high level (level 30 or up) Pokémon that knows False Swipe. False Swipe is a 40 power move that will never knock out an opponent (it will instead leave them with 1 HP), perfect for catching Pokémon. Grovyle or Sceptile. If you started with Treecko, level it up to level 53 as a Grovyle or 59 as a Sceptile to learn False Swipe. You can also TM it False Swipe. Nincada. Catch a Nincada on Route 116 (uncommon) and train it to level 25 without evolving to teach it False Swipe. Trade. Alternatively, trade a level 46 Farfetch'd, level 33 Cubone, level 39 Marowak, level 16 Scyther or Scizor, a Smeargle that Sketched False Swipe, or a level 55 Zangoose that knows False Swipe.
Bring a high level (Level 30 or up) Pokémon that knows a sleep inducing move. When Relicanth is asleep, it is much easier to capture. An alternative (though less effective), is paralysis inducing moves. Ralts, Kirlia and Gardevoir. Catch a Ralts on Route 102 (rare) and train it to level 41 as a Ralts, level 47 as a Kirlia or level 51 as a Gardevoir, to teach it Hypnosis. Spinda. Catch a Spinda on Route 113 (common) and train it to level 23, to teach it Hypnosis. Oddish or Gloom. Catch an Oddish or Gloom on Route 110, 117, 119, 120, 121 or 123, or at the Safari Zone, and train it to level 18 to teach it Sleep Powder.
Shroomish or Breloom. Train it to Level 54 as a Shroomish to teach it Spore.
Dive underwater at Route 126 and wander through the seaweed until you find a Relicanth. They are very rare (1/20 chance of appearing), so patience is necessary.
Weaken Relicanth, without knocking it out, using False Swipe. Use other moves if necessary, but when Relicanth gets weak, make sure you use False Swipe, so as to not accidentally knock it out.
Induce Relicanth with Sleep or Paralysis. Sleep is more effective than Paralysis.
Throw Dive Balls until you catch Relicanth. This can take a while, as Relicanth has a very low catch rate, similar to some legendaries. If Relicanth wakes up, remember to put it back to sleep.
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