C550 ON ILS RWY 2L APCH TO DPA IS UNABLE TO MAINTAIN ALT DURING ILS ATTEMPT DUE TO WX.

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OUR CITATION JET WAS BEING VECTORED FOR THE ILS RWY 2L AT DPA ARPT BY CHICAGO APCH. AFTER RECEIVING A CLRNC TO SHOOT THE APCH; WE ENCOUNTERED SEVERE CONVECTIVE ACTIVITY OUTSIDE THE FAF AND WERE UNABLE TO MAINTAIN ALT. WE ENTERED HVY RAIN AND EXPERIENCED MODERATE TO OCCASIONALLY SEVERE TURB. A SUDDEN INCREASE IN AIRSPD WAS NOTED AND THE PLANE BEGAN TO CLB UNCONTROLLABLY EVEN THOUGH THE APPROPRIATE PITCH AND PWR CHANGES WERE BEING MADE TO CORRECT THE CLB. ATC ASKED US WHY WE CLBED TO 4000 FT; AND I RESPONDED THAT WE WERE IN A CELL. ATC DIRECTED US TO MAINTAIN 3000 FT; AND I INFORMED THE CTLR THAT WE WERE UNABLE TO CTL THE ACFT AND REQUESTED AN IMMEDIATE TURN. WE WERE GIVEN A TURN OFF COURSE AND REQUESTED ANOTHER 90 DEG TURN TO EXIT THE WX. WE WERE EVENTUALLY VECTORED TO THE W SIDE OF THE LINE OF WX AND WERE ABLE TO SUCCESSFULLY SHOOT ANOTHER APCH TO RWY 2L. ATC DID NOT INFORM US OF ANY TFC CONFLICT AND NONE WAS NOTED ON THE TCASII. CONTRIBUTING FACTORS: ALTHOUGH WE HAD ONBOARD RADAR WE WERE UNAWARE OF THE SEVERITY OF THE LINE OF WX ENCROACHING ON THE FINAL APCH COURSE. IT APPEARED TO BE A VERY NARROW BAND; BUT THE SCALE OF THE RADAR DISPLAY PROBABLY SHOULD HAVE BEEN ON A LOWER SETTING TO BETTER DEPICT THE CELL AND ITS POS RELATIVE TO THE ILS FINAL APCH COURSE (THE DISPLAY MAY HAVE BEEN SET ON 25 MI INSTEAD OF 12.5). ALSO; ATC GAVE NO INDICATION OF ANY CONVECTIVE ACTIVITY ALONG THE FINAL APCH COURSE.