Get in, hit hard, and get out. If that's not how you're using Elgyem, you're doing it wrong. Elgyem is bulky enough to come in on weaker Pokemon, such as Bronzor, Foongus, and Natu, as well as Fighting-types such as Timburr, and safely spam its high powered, Analytic-boosted attacks. Speaking of Analytic, it is Elgyem's most defining attribute and what sets it apart from other Psychic-types, especially in tandem with its base 85 Special Attack. There aren't many Pokemon in LC that can avoid a 2HKO from Elgyem, provided Elgyem uses the right combination of attacks. This of course means that when you're using Elgyem, prediction is key. However, if you can land the correct attacks, Elgyem becomes a beast. Psychic is an odd move to spam in a tier where Pawniard, Bronzor, and Ferroseed are ubiquitous, but it is Elgyem's most powerful attack against most Pokemon. Thunderbolt or Hidden Power will hit the aforementioned Pokemon hard though, so they usually aren't a problem with decent prediction. With Recover, Elgyem can stay in the game for the long haul against teams where it is able to switch in multiple times.
For a Pokemon that is meant to utilize Analytic to its full effect, running a Modest nature with Speed EVs might seem strange at first; with the given spread, Elgyem will hit a Speed of 11. This allows Elgyem to outspeed many slower walls, so it can hit them with an Analytic-boosted attack while they switch in and then finish them off before they can move. This Speed investment also gives Elgyem an easier time against specific Fighting-types, mainly Croagunk (and to an extent Timburr). With its Special Attack maxed out, Elgyem hits 19 Special Attack, and with what is essentially a free Life Orb boost to most of its attacks, Elgyem is one of the hardest hitting Pokemon in LC. Even with just 76 HP EVs, Elgyem will have the bulk to not only come in on weaker Pokemon but also take some STAB super effective moves at full health in a pinch.
The above combination of moves is usually the best, but Elgyem does have a couple of other options. Shadow Ball is Elgyem's strongest attack against Misdreavus and opposing Psychic-types, and although it won't OHKO Misdreavus, it comes fairly close. Calm Mind can also be used over Thunderbolt or Hidden Power. This lets Elgyem set up on many weaker Pokemon and beat bulkier Pokemon it would otherwise not be able to, such as Lileep. Alternatively, Elgyem can run Charge Beam instead of Thunderbolt, allowing it to attack and set up simultaneously. However, without the Special Defense boosts Calm Mind provides, Elgyem will be even easier to revenge kill, and the initial drop in power is noticeable. If the EV spread irks you, a bulkier spread of 236 HP / 240 SpA with a Quiet nature is an option. This spread allows Elgyem to survive almost any attack if it's close to full health, such as Sucker Punch from Eviolite Pawniard or Shadow Ball from Misdreavus. If this spread is chosen, make sure to set the Speed IV to 2, which minimizes Elgyem's Speed for Analytic while still keeping Hidden Power Fighting.
This little alien requires a bit of support to bring out its true potential. Like every Pokemon, Elgyem greatly appreciates entry hazard support, as in many instances it's the difference between nabbing or missing an OHKO or 2HKO. Thus, Pokemon such as Ferroseed, Lileep, and Tirtouga are good partners, as they can set up Spikes and Stealth Rock respectively. Elgyem works well with these Pokemon in particular as Ferroseed and Lileep cover the Water-types that Elgyem has a neutral match-up with, at best, while Tirtouga is a very solid Vullaby check. Additionally, Elgyem helps to cover some of the Fighting-types that threaten these Pokemon.
Elgyem will often come in on bulky Pokemon that carry status moves, so a cleric is much appreciated. Chinchou is the obvious choice, and also helps against Vullaby to boot. Lickitung is an option as well, and can use Knock Off to make Elgyem hit even harder by removing Eviolites. Pokemon that lure in the bulky Pokemon that Elgyem can come in on are almost mandatory. Chinchou helps out in this regard, drawing in weaker Pokemon such as Foongus and Ferroseed. Ferroseed is again a great partner for this reason, as it is bait for Natu and Fighting-types.
Because three of Elgyem's biggest checks—Pawniard, Vullaby, and Misdreavus—are all top threats themselves, it's important to include teammates that can deal with them, especially because some other helpful teammates will surely aggravate these weaknesses. The aforementioned Chinchou, Tirtouga, and Ferroseed all work well (Chinchou and Tirtouga for Vullaby, Ferroseed for Misdreavus), as well as helping out Elgyem in other ways. Some other helpful teammates include Mienfoo, which can check Porygon and use either U-turn to get Elgyem in safely or Knock Off to remove Eviolite from opposing Pokemon, and Archen, which can check Vullaby and either set up Stealth Rock or provide U-turn support.