CARJ CREW; FINDING THEMSELVES HIGH AT THE FAF; EXECUTED A MISSED APCH.

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DUE TO WX ON THE APCH TO RWY 31C AT MDW; WE WERE VECTORED TO THE LOC FOR RWY 31C INSIDE RUNTS INTXN. DUE TO 66 KT XWINDS; WE DID NOT INTERCEPT THE LOC UNTIL JUST BEFORE FAF KEDZI. THE GS WAS FULLY DEFLECTED AT KEDZI; SO WE EXECUTED A MISSED APCH. ON THE MISSED APCH WE SAW ON THE EFIS THAT THE PUBLISHED MISSED APCH PROC STEERED US DIRECTLY INTO A TSTM (WE HAD SEEN LIGHTNING IN IT). WE WERE UNABLE TO MAKE THE L TURN AND I KNEW BUILDINGS WERE TO THE R. WE PROCEEDED ON RWY HDG AND WERE TOLD TO CONTACT DEP; FOR MISSED APCH INSTRUCTIONS (AFTER WE TOLD THEM WE COULD NOT EXECUTE THE PUBLISHED MISS). WE TRIED TO CONTACT DEP BUT DUE TO THE VOLUME OF RADIO TFC THE PNF COULD NOT GET A WORD IN. I PROCEEDED TO THE MSA (3400 FT). THE ORD CTLRS FINALLY GOT TO US AND TOLD US TO DSND FOR TFC TO 3000 FT WITHOUT DELAY. I STARTED DOWN; BUT WE WERE IN BAD TURB AND CTLING THE ACFT BECAME DIFFICULT. I PWRED BACK AND ESTABLISHED A DSCNT. WHEN I TURNED ON THE AUTOPLT AT 3200 FT (DSNDING AT 800 FPM); THE NOSE PITCHED UP AND THE PLANE WENT TO 3400 FT BEFORE I COULD DISCONNECT IT AND RE-ESTABLISH A DSCNT. AFTER I ESTABLISHED THE ACFT AT 3000 FT ON THE NEW HDG; I WAS ABLE TO RE-ACTIVATE AUTOPLT. AFTER THAT; IT WORKED PROPERLY. THE BAD TURB MADE COMPLYING WITH ATC INSTRUCTIONS HARD AND THE CONGESTED DEP FREQ COMPOUNDED THE PROB. THE AUTOPLT MADE THINGS WORSE INITIALLY; BUT WAS USEFUL AFTER WE GOT INTO SMOOTHER AIR AND IT STARTED WORKING.