A C-560 FLT CREW FAILED TO PROPERLY BRIEF THE INST APCH WHICH LED TO CONFUSION AT MISSED APCH POINT AND AN ALTDEV DURING THE MISSED APCH.

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I AM A CPR PLT AND WAS PERFORMING SIC DUTIES ON BOARD ONE OF OUR COMPANY ACFT ON A PART 91 OP FLYING TO KISM. UPON ARR TO THE ORLANDO AREA; THERE WERE TSTMS AND RAIN SHOWERS IN THE AREA. I RECEIVED THE KISM AWOS ABOUT 10 MINS BEFORE OUR ARR AND THE WX WAS RPTED AS LIGHT WINDS FROM THE SW; VISIBILITY 4 MI WITH MIST; AND 1400 FT OVCST. ORLANDO APCH TOLD US TO EXPECT THE NDB TO RWY 15. OUR CHKLIST CALLED FOR AN APCH BRIEFING; WHICH THE PIC AND I COMPLETED. ONCE WE GOT WITHIN 10-15 MI OF THE ARPT; THE CTLR STATED THAT THERE WAS 'LEVEL 3' WX ON THE APCH COURSE TO OUR INTENDED RWY; WHICH WE VERIFIED WITH OUR WX RADAR. AFTER A SHORT DISCUSSION; THE PIC AND I AGREED ON THE GPS TO RWY 6; CIRCLE TO RWY 33. WE HAD JUST ENOUGH TIME TO SET UP THE AVIONICS FOR THAT APCH BEFORE WE INTERCEPTED THE IAP FOR THE APCH. WE DID NOT FULLY BRIEF THE APCH; MOST NOTABLY THE MISSED APCH PROC. THE TWR CTLR TOLD US TO CALL THE ARPT IN SIGHT AND PLAN ON A L BASE ENTRY TO RWY 33. UPON REACHING MDA FOR CIRCLING MINIMUMS; THE PIC AND I WERE BOTH SURPRISED TO STILL BE IN THE CLOUDS. SHORTLY AFTER THAT; WE REACHED THE MISSED APCH POINT AND BROKE OUT AT THE SAME TIME. THE PIC INITIALLY BEGAN A SHALLOW CLB; BUT THEN CALLED THE ARPT IN SIGHT. HOWEVER; WE WERE TOO CLOSE TO THE ARPT FOR A L BASE ENTRY TO RWY 33; SO WE ASKED THE TWR IF A R CIRCLE TO THE SE OF THE ARPT WOULD BE APPROVED. AFTER A SHORT DELAY; THE CTLR RESPONDED 'THAT WOULD BE FINE; RPT A R BASE TO RWY 33.' IMMEDIATELY AFTER THAT; WE FLEW INTO IMC CONDITIONS AND LOST SIGHT OF THE ARPT. THE PIC CALLED MISSED APCH; AND CONTINUED HIS CLB. I RADIOED THE TWR OF OUR MISSED APCH. THE TWR TOLD US TO IMMEDIATELY TURN TO A HDG OF 330 DEGS (ALMOST A 100 DEG L TURN FROM OUR CURRENT HDG). WE INITIATED THE TURN. THE PIC THEN ASKED ME WHAT THE MISSED APCH PROCS WERE. I STARTED TO READ THEM OUT WHILE TRYING TO PROGRAM THE FMS TO THE MISSED APCH POINT (OUR FMS DOES NOT DO THIS STEP AUTOMATICALLY; AS SOME DO). THE PIC THEN ASKED ME WHAT ALT TO CLB TO; TO WHICH I REPLIED '2000 FT.' AT THAT POINT THE ACFT WAS AT 2800 FT AND THE PIC LOWERED THE NOSE TO LEVEL AT 3000 FT. SHORTLY AFTER THAT; WE WERE ASKED BY ORLANDO APCH IF ISM TWR ASSIGNED US 3000 FT. I REPLIED 'NEGATIVE.' APCH THEN ASSIGNED US A CLB AND WE REQUESTED TO DIVERT TO KMCO. WE ENDED UP LNDG AT KMCO. CONCLUSION: BOTH THE PIC AND MYSELF WERE UPSET AT OURSELVES BY THIS EVENT. IT ALL BEGAN WITH RAPIDLY DETERIORATING WX; PRESSURING US TO 'RUSH' OUR NEXT APCH BRIEFING. WE DID NOT EXPECT THE WX TO BE A FACTOR ONCE WE SWITCHED TO THE SECOND APCH AND WE WERE NOT PREPARED FOR THE MISSED APCH. ALSO CONTRIBUTING WAS OUR INDECISIVENESS ON WHAT TO DO ONCE WE BROKE OUT OF THE CLOUDS. SINCE WE WERE IN NO POS TO MAKE THE PLANNED L BASE TO RWY 33; WE SHOULD HAVE EXECUTED THE MISSED APPROACH FROM THAT POINT; RATHER THAN TRYING TO QUICKLY CHANGE OUR PLAN AND CIRCLE THE OTHER WAY. WE BOTH GOT 'BEHIND' THE AIRPLANE ONCE WE SWITCHED TO THE OTHER APCH AND WE SHOULD HAVE REQUESTED A VECTOR OR SOMETHING TO ALLOW US TIME TO PROPERLY SET UP AND BRIEF THE APCH. THIS WAS A GOOD LESSON ON HOW THE 'ERROR CHAIN' CAN RUN WILD. WE WERE LUCKY THAT WE DIDN'T CAUSE A CONFLICT RESULTING FROM OUR MISTAKE. WE ARE RARELY FORCED TO EXECUTE A MISSED APCH; AND USUALLY IF THAT HAPPENS; WE ARE SOMEWHAT PREPARED FOR IT. THIS TIME; THE WX RAPIDLY DETERIORATED TO FROM MVFR CONDITIONS TO BELOW CIRCLING MINIMUMS WITHIN A 15-MIN PERIOD AND IT CAUGHT US BOTH OFF GUARD.