C550 DEP FROM DPA EXCEEDS ALT CLRNC WHILE TRYING TO AVOID WX.

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OUR CITATION JET HAD DEPARTED DUPAGE ARPT HDG ABOUT 200 DEGS WITH A CLRNC TO CLB TO 6000 FT. DURING THE CLB; ATC DIRECTED US TO TURN L TO 140 DEGS. IN AN EFFORT TO AVOID THE WX DEPICTED ON THE RADAR SCREEN; AND BASED ON OUR EXPERIENCE FROM A SEVERE ENCOUNTER WITH THE SAME WX SYS ONLY 1 HR PRIOR TO THIS FLT; WE ASKED ATC IF WE COULD MAINTAIN OUR PRESENT HDG. ATC FLATLY SAID 'NO' AND OFFERED NO OTHER OPTIONS. AS THE PF; I BEGAN A SLOW TURN TO 140 DEGS WHILE TRYING TO PICK THE BEST RTE THROUGH THE RAIN SHOWERS DEPICTED ON THE RADAR. I WAS LISTENING TO THE PNF'S CONVERSATIONS WITH ATC AND WAS TRYING TO CHOOSE A RTE THROUGH THE WX AND NEGLECTED TO LEVEL OFF AT 6000 FT. WE EXCEEDED THE ALT BY ABOUT 700 FT BEFORE THE MISTAKE WAS CORRECTED. ATC DID NOT MENTION ANY TFC CONFLICT AND NONE WAS NOTED ON OUR TCASII. CONTRIBUTING FACTORS: DURING THE SUBSEQUENT PORTION OF THE FLT; IT WAS NOTED THAT THE AURAL PORTION OF THE ALT ALERTER WAS NOT FUNCTIONING. A TEST OF THE WARNING SYS WAS PERFORMED AT THE DEST ARPT; AND THE FAILURE WAS CONFIRMED. THE ALT ALERTER WAS PLACARDED INOP ACCORDING TO THE ACFT'S MEL. HAD THE AURAL PORTION BEEN WORKING I WOULD HAVE RECEIVED A WARNING AT 5000 FT THAT I WAS APCHING MY ASSIGNED ALT EVEN THOUGH I WAS DISTR BY LOOKING AT THE RADAR SCREEN. I WOULD HAVE ALSO RECEIVED ANOTHER WARNING AFTER PASSING THE 6000 FT LEVEL. THE ACFT WAS BEING HAND FLOWN AT THE TIME. IF IT HAD BEEN PLACED ON THE AUTOPLT SOONER; THE PLANE SHOULD HAVE LEVELED OFF AT THE PRESCRIBED ALT DESPITE THE FACT THAT THE AURAL WARNING PORTION OF THE ALT ALERTER HAD FAILED. UNFORTUNATELY; THE AUTOPLT DIDN'T HANDLE THE TURB WELL ON THE INBOUND FLT (AND IN FACT HAD DISCONNECTED DUE TO ITS INABILITY TO MAINTAIN ALT); SO I DECIDED TO HAND FLY THE PLANE THROUGH THE WX. AS THE PF; I ALLOWED MYSELF TO BE DISTR FROM THE PRIMARY FUNCTION OF FLYING THE PLANE. I BELIEVE IT WAS A DIRECT RESULT OF WANTING TO AVOID A REPEAT EXPERIENCE OF THE ENCOUNTER WITH SEVERE CONVECTIVE ACTIVITY ON THE FLT INTO DUPAGE 1 HR OR SO EARLIER. ANOTHER CONTRIBUTING FACTOR WAS ATC'S TOTAL UNWILLINGNESS TO WORK WITH US OR OFFER ANY OTHER OPTIONS THAN A TURN INTO THE WX. I WASN'T AWARE OF ANY OTHER ACFT IN THE IMMEDIATE VICINITY; AND ATC DIDN'T CITE CONFLICTING TFC AS A REASON FOR THE TURN. THEY SIMPLY SAID THAT THEY WERE ATC AND WE HAD TO DO WHAT THEY SAID.