BE30 CREW HAD ALT OVERSHOOT ON CLIMBOUT AVOIDING WAKE TURB FROM A DEPARTING ACR JET.

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ON THIS DAY; I THE CO-PLT HAD JUST COMPLETED A 4.7 HR LEG AS PNF. WE HAD STOPPED IN TPA TO CLR CUSTOMS AND TAKE ON FUEL FOR OUR FINAL LEG. ON THIS FINAL LEG I WAS THE PF; WITH MY CAPT HANDLING THE RADIOS AND OTHER NORMAL PNF DUTIES. AFTER RECEIVING OUR INITIAL DEP CLRNC; FROM RWY 18L; UP TO 1600 FT AND R TO 310 DEGS; WE TAXIED OUT AND DID NORMAL PRE-TKOF CHKLISTS. AT THE THRESHOLD I CALLED THE TWR AND WAS INSTRUCTED TO HOLD WHILE A HEAVY JET OFF RWY 18R ACHIEVED 5 MILE SEPARATION. AFTER WAITING WE WERE CLRED FOR TAKEOFF; WITH A CAUTION TO BE AWARE OF WAKE TURB; AS WE WOULD BE CROSSING TO OUTBOARD PATH OF THE HEAVY JET OFF RWY 18R. WE EXECUTED A NORMAL TAKEOFF AND INITIAL CLIMB; AND AT APPROX 400 FT I BEGAN A R TURN TO 310 DEGS. MY CAPT ADVISED ME TO CLIMB RAPIDLY TO ASSURE SEPARATION FROM THE AREA OF WAKE TURBULENCE. MY CONCERN WITH MAINTAINING PROPER CTL WHILE EXECUTING THE TURN TO 310 DEGS; AND WHILE CLIMBING RAPIDLY AT A LOWER THAN NORMAL AIRSPEED; CAUSED ME TO BE DISTRACTED. I FORGOT THAT I HAD ONLY BEEN CLRED TO 1600 FT. AT APPROX 2400 FT THE TWR CTLR INSTRUCTED US TO GO TO DEP FREQ. MY CAPT CHKED IN WITH THE DEP CTLR; STATING THAT WE WERE AT 2400 FT AND CLIMBING. AT THAT POINT WE REALIZED THAT I HAD CLIMBED WELL BEYOND OUR CLRED ALTITUDE. WE WERE INSTRUCTED TO RAPIDLY RETURN TO 1600 FT; WHICH I DID. WE HAD NO FURTHER PROBS.