Level 100 Statistics (see level 5, 50, 100)
The lack of Tail Glow condemns Illumise to never be used. It has better stats to use Tail Glow than Volbeat, but Game Freak decided that Illumise shouldn't learn it, which is really too bad, as it would be a decent Baton Passer in UU otherwise.
It's too bad Illumise can't learn Tail Glow, as it can use that much better than Volbeat thanks to a higher Special Attack stat. Oh well. Sub up as much as possible, then Baton Pass out when you cannot kill. You can Substitute down to activate a Salac Berry, which gives you a chance at a sweep if you pass to something with high Speed.
If you go with Thunder Wave, use the EVs given, as Speed is irrelevant with a Thunder Wave. If you use Light Screen, you might want to invest a few EVs there.
Moonlight works for some healing reasons if you think you can survive a hit and don't want Wish.
Illumise will generally want 270 Speed if she has Ice Punch, as this beats Adamant Gligar. This requires 160 EVs and a +Speed nature. If Illumise doesn't have moves to hit Gligar hard, this is much less important, as most Pokémon Illumise needs to outspeed don't invest in Speed EVs and have a lower base Speed (such as Solrock), so a minimal investment is all that is needed.
Checks and Counters
UU counters: Gligar can defeat Illumise if it avoids Ice Punch. Fearow can switch into Wish and Light Screen, and makes a decent counter with its superior Speed and STAB on Drill Peck, but it doesn't like switching into Thunderbolt or Ice Punch (although Ice Punch won't do enough to KO). Hitmonlee will usually outspeed Illumise, and can OHKO with Rock Slide; Hitmonlee has the Special Defense needed to take Thunderbolt and Ice Punch. Kangaskhan just absorbs any of Illumise's weak hits and retaliates with STAB Return. Basically any heavy hitter that is either faster than Illumise or isn't weak to one of her moves (and even if it is weak, Illumise probably won't do that much, except maybe with Thunderbolt) can counter her.