A King Air Pilot reported encountering wake vortex from a B737 departing LAX; that distraction and others leading to an altitude overshoot at 7000 ft.

Narrative

We were departing Los Angeles and we were given a position and hold clearance for Runway 25R. We had an aircraft caution light come on and immediately go out while we waited to go. The aircraft was normal and we received our clearance for takeoff. There had been a 737 takeoff prior to us and the wind was calm. At about 1200 FT we got into the wake of the other aircraft and were IMC using full control inputs to maintain wings level. We immediately increased our climb rate and asked for relief on the departure SID which has two at or below altitude restrictions. We were granted an altitude of 6000 FT. We were now VMC; but unsure if the other aircraft wake was still a factor for us. We were cleared to 7000 FT and given a clearance to turn left direct to the LAX VOR. At this point the climb was decreased; and we started our turn. Still hand flying and with numerous distractions; the aircraft level off at 7000 FT was late and we flew through the altitude by almost 400 FT before immediately correcting to the given altitude of 7000 FT. The controller immediately gave us a climb to 13000 FT. Nothing was said by the controller; and we had no TCAS traffic. I think that looking back at this situation I could have done things better. The wake encounter flustered me a bit and allowed me to be distracted away from 100% flying the aircraft. The Captain was also task saturated. We have since started asking the tower for relief on the SID prior to takeoff if behind heavier aircraft. I have a renewed vigilance for altitude parameters.