Unfortunately there's only one chance to get Absol in both games and it's fairly rare. However, the Absolite is available in the Safari Zone on the very next route which gives Absol an immediate buff that it can ride through the rest of the game. It's handy against Team Aqua/Magma and the seventh Gym, has a ridiculous movepool, and Magic Bounce gives it a multitude of chances for setting up and sweeping. If you've got space on the team it's worth a slot, especially as a lead.
Courtney/Matt (Mt. Pyre): A Night Slash from Mega Absol should deal with Camerupt, but regular Absol may not make the cut. If you're Mega, you can outspeed Sharpedo and KO it with Thunderbolt or Thunder, but otherwise regular Absol will be outsped, and that can be dangerous for its frailty.
Rival (Lilycove City): Absol doesn't have a lot of advantages in this fight. The fully-evolved starters all pack a punch and Absol's STAB doesn't help it get the necessary OHKOs. Not to mention Breloom and Blaziken will OHKO it with a fighting move. Generally it's best to stay out of this fight unless you want to take on Wailord, Magcargo, or Swellow because you don't have better options. However, you can get Absolite in the Safari Zone before this battle, so it might be fun to use, especially if at level 37 or higher already; it gets Psycho Cut, which actually warrants a chance against the Fighting-types.
Courtney/Matt (Lilycove Hideout): Same as the previous fight. Do it if Mega, don't do it if not.
Gym #7 - Tate and Liza (Mossdeep City, Psychic-type): Fortunately Absol is not so limited in this battle. If you've got it up to level, have at this gym with no fear. Night Slash will OHKO Lunatone and has about a 50% chance on Solrock as well (given same levels), and there's no real fear from Rock Slide as even a crit won't OHKO Mega Absol. With a good partner this could easily be a one turn battle.
Maxie/Archie (Seafloor Cavern): Mega Absol is great for repelling Mightyena's Taunt/Scary Face and setting up for a sweep here. Mightyena can't touch it, so feel free to get to +6 attack with Sword Dance and destroy everyone. Or give someone else on your team a chance, if you want.
Primal Groudon (Cave of Origin, Omega Ruby only): If you don't want even Mega Absol to be destroyed, then maybe, just maybe, sit this one out.
Primal Kyogre (Cave of Origin, Alpha Sapphire only): See Primal Groudon, except on the special side. Even Thunder will not do a lot, so just don't bother.
Gym #8 - Wallace (Sootopolis City, Water-type): Luvdisc, again, is prime set-up material for Mega Absol, just beware of Draining Kiss doing too much damage. It's fairy easy to get to +6 again and destroy Wallace where he stands, thanks to that 115 base speed.
Wally (Victory Road): A sweep is easily possible here as well, as long as you can stay healthy while building up Sword Dances against Altaria. Aerial Ace is the perfect coverage move for this battle, hitting Roselia and Gallade for OHKOs once you've boosted. Otherwise, Night Slash will serve you fine. The only thing to watch out for as the inevitable shot from Magneton due to Sturdy.
Elite Four Sidney (Ever Grande City, Dark-type): Thought Absol would be useless in a Dark-only battle? Think again. You find the TM for X-Scissor in Victory Road. Slap it on Absol and watch the destruction commence. Per the usual, boost to +6 while watching your health and have at it. Mega Absol has no match in speed in the E4, so you shouldn't have to worry about getting outsped unless you're underleveled.
Elite Four Phoebe (Ever Grande City, Ghost-type): LOL go get 'em.
Elite Four Glacia (Ever Grande City, Ice-type): It's getting a little silly at this point, but you can remove Aerial Ace or X-Scissor for Rock Slide, or even Rock Tomb and let the sweeping commence as usual.
Elite Four Drake (Ever Grande City, Dragon-type): Drake finally rains on Absol's parade. Altaria's Moonblast simply won't let you get set up for a sweep here. While there is minimal chance you get OHKO'd, you won't be able to keep the pressure on (unless you were astoundingly lucky and got Play Rough). No one else carries a super-effective move, but switching Absol in is tough with those unfortunate defenses. Absol is better left on the sidelines for this one battle, unless you can Mega Evolve and also have Ice Beam; in that case, Drake will actually be pretty easy, as most of his team is twice as weak to Ice and can be easily OHKOed by an Ice move. Kingdra is the only exception.
Champion Steven (Ever Grande City, Steel-type): Thanks to the Steel-type nerf that makes Dark-type moves neutral, Absol can once again go for gold. There is one issue though...both Skarmory and Aggron have Sturdy. Skarmory isn't a huge deal as it can't dish out too much damage, but Aggron can make short work of Absol with STAB Stone Edge. Soooo if you happen to have caught an Absol with Baton Pass you can pass those boosts onto someone else, or you can sit it out against Aggron. Unfortunately losing the boosts makes it hard to switch back in, but depending on level discrepancies you may be able to get back in against Cradily if you haven't already KOd it.
You will catch Absol at level 27 with Sword Dance, Slash, Double Team, and Bite, an already potent set. Unless you really want Quick Attack, there's no reason to reteach any old moves. Night Slash will quickly replace Bite at 29 and stick with you through the rest of the game. Detect isn't worth a slot, but Psycho Cut at 37 is another crit-happy move that covers Fighting-types for Absol. Me First is useless, and Sucker Punch, while deadly competitively, has minimal usefulness at this stage in the game where no one is using priority moves back at you. Everything else in the late moveset is garbage unless you want to get gimicky with Perish Song.
Absol boasts one of the largest movesets in the entire series, thank in large part to TMs. Hone Claws can be used in place of Sword Dance, but generally the latter is better. Return is powerful but doesn't offer extra coverage. Aerial Ace provides possibly the best coverage, hitting Bug and Fighting super effectively, leaving Absol only vulnerable to Fairy. Shadow Claw is yet another crit-getter, but covers almost the same thing as Night Slash but without STAB. Stone-Edge and Rock Slide situationally can be great coverage moves, as can X-Scissor. And that's just the physical side. Thanks to its Mega Evolution, Absol can use Special moves effectively as well, although I don't find it necessary. Calm Mind, Ice Beam or Blizzard, Thunderbolt or Thunder, Flamethrower or Fire Blast, and Dark Pulse are all options on the Special side if you see the need, but it is much easier to buff the already superior Physical side. Lastly, Will-O-Wisp and Thunder Wave are both crippling status moves that Absol can wield. If you happen to catch one with an egg move, there are a few nice options. Megahorn is superior to X-Scissor in power, but the accuracy probably isn't worth it. Play Rough is a nice coverage move, especially for Drake and the post game. Baton Pass is a wonderful move to pass on those Sword Dance boosts when Absol is countered.
Recommended moveset: It's hard to tie anything down with so many options, but generally 1 set up move + 3 attacks, one being Night Slash
Minimum stats are calculated with 0 EVs, IVs of 0, and a hindering nature, if applicable.
Maximum stats are calculated with 252 EVs, IVs of 31, and a helpful nature, if applicable.
What nature do I want? Anything that lowers Special Attack is my suggestion, unless you intend to be a mixed attacker. Attack raising natures are good, but if it's at the cost of one of Absol's defenses you'll need to be careful. Speed raising is also ok, but as a mega you will already outspeed almost everything in the game with 115 base speed.
How good is Absol in a Nuzlocke? Absol is a lot of fun and at times game-breakingly easy to use. Just be careful on switch ins with those low defenses.