Level 100 Statistics (see level 5, 50, 100)
There is no denying the cool factor of using Relicanth. It is usable in OU, UU, and NU but really shines in the latter. Its Defense is typically underestimated by those not familiar with it, and it will survive most non-Choice Band-boosted super effective moves with relative ease. Relicanth has the best physical defense available in NU which might sound like damning with feint praise, but it really does its job well. Highly recommended.
With its massive Defense, Sleep Talk enhances Relicanth's survivability and lets it continue to wall. Typically this is the set you'll want to use outside of Relicanth's home metagame, Neverused, and also makes it one of the premier defenders inside of NU. Feel free to use Double-Edge over either Rock Slide or Earthquake depending on your team's need; just make sure you change its ability to Rock Head if you choose to do so.
Similar to Choice Band Sudowoodo in NU, without Explosion. The advantages of using Relicanth for this are the obviously better defenses and the surprise factor. Your opponent likely will not expect this kind of attacking power from a notoriously defensive Pokemon. The last slot is generally filler and can be used to hit Grass-types harder or to keep attacking if Relicanth gets put to sleep. You probably won't be running into a lot of sleep in NU, and most sleep inducers are Grass-types, which Relicanth does not want to fight against anyway, but it can be a decent move to let you switch into a predicted sleep and come right back out, so you can still attack with it later.
Yawn is a nice pseudo-phazing move. Psych Up can help you take on Curse and Baton Pass users. Beyond that, there isn't much. Relicanth also learns Calm Mind and the typical Water and Ice moves to go along with it, but any type of special attacking on Relicanth is mostly novelty.
For any Resting version you'll want to invest heavily in HP and Defense, dumping the remaining EVs into Attack. You should tailor the EVs to the specific metagame you plan to use Relicanth in. Make sure it survives something a certain number of times or can kill something that you want it to counter and go from there. For the Choice Band version, the only question is if you want to outspeed anything close to the 146 Speed Relicanth has with no EVs and a neutral nature. Use enough to outspeed it, max Attack, and use the rest in HP.
Checks and Counters
There isn't a whole lot to worry Relicanth here. Whiscash can give it pause, but there is no real 100% counter in NU. Plusle might scare it off after a kill but doesn't like Rock Slide or Earthquake if it tries to switch in. Likewise with Sunflora and Huntail.