Well if it's your favourite Pokemon you might as well use it.
Elgyem's growth rate is in the Medium-fast one (requires 1 million EXP at level 100). The same holds true for Beheeyem (Elgyem evolves to Beheeyem at level 42), so in the end it's quite a fast Pokemon to train, and won't take as much work as other Psychic Pokemon (eg. reuniclus) would. These are the different growth rates btw
Elgyem's stats are 55/55/55/85/55/30 (HP, Atk, Def, SAtk, SDef, Spe)
Beheeyem's stats are 75/75/75/125/95/40 (Same format)
As you can see, it's speed is downright horrible, with strong SpA and passable bulk.
Unfortunately, the only moves both of them can learn by levelling up are all normal and psychic attacks. Not much variety is there =/, however they do get Calm Mind and Recover which are pretty decent moves. Additionally, alot of the moves are more support based, so Beeheyem is a nice Doubles Pokemon or a Trick Room attacker, but otherwise not really good. However, TMs do add a little variety to their movesets, so they can still be used.
Beheeyem/Elgyem have quite standard normal abilities. being Synchronise and Telepathy. Synchronise is the most useful out of the two in my opinion, since Telepathy only really works in double battles, and Single Battles are the most common. Their Dream World ability, Analytic on the other hand is excellent - +30% power if they move last, which they are probably goign to do considering their abysmal speed
Anyway, in the end I say Beheeyem/Elgyem are averagish kind of psychic types, and aren't too bad but aren't all that good either. If you want, go ahead and use them. A movepool that could work with them is;
~Calm Mind (learned)
~ Shadow Ball/Recover (TM/learned)
~ Thunderbolt (TM)
~ Psychic (TM)
You'll prefer a special attack boosting nature (best to be modest) and like I said before, the better ability would probably be Synchronise, under the assumption you can't get Analytic
Obviously they aren't goign to outspeed much, so you'll need to get used to hitting after. Alternatively run a Trick Room set (TM 92) on Beheeyem which lets you take advantage of it's low speed, though stick with the above set if you have Analytic.
Personally, there are better psychic types out there like Reuniclus (Slower, but better bulk/ability and same SpA ) and Gothitelle (Better bulk, Awesome Dream World ability if you can get it, better speed). They do take longer to train however (medium-slow group for EXP). My top suggestion for a Psychic type, though harder to get, would be Espeon - Higher speed, higher special attack and only a little worse bulk than Beheeyem. It also takes the same amount of time to train. However if you want a fairly quick Pokemon to train that is relatively easy to obtain, Beheeyem is an acceptable choice. However, like I said before since it's your favourite Pokemon that probably outweighs everything so why not go ahead?
Also here's Marriland's rate of Elgyem/Beheeyem in B/W2
You can check here to get Beheeyem's movepool and that stuff.