ACR X NON ADHERENCE TO ATC CLRNC CLB THROUGH ASSIGNED ALT. TCASII TA FLC DISTR. PLTDEV.

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ACR X ON CLBOUT FROM ORD; WE WERE ON A HDG OF 040 DEG AND PASSING THROUGH APPROX 4700 FT WHEN WE RECEIVED CLRNC TO CLB TO 6000 FT AND TURN TO 140 DEG HDG. ROLLING IN 1000 FPM PITCH ON FLT GUIDANCE PANEL WE STARTED A TURN AND A CLB AT APPROX 5800 FT RECEIVED A TA ON TCASII AND VERBAL TFC; TFC ALERT. LOOKING DOWN AT TCASII MOMENTARILY AND THEN OUTSIDE TO SEE POTENTIAL TFC CONFLICT; ACFT PASSED THROUGH 6000 FT. AFTER SPOTTING TFC AND RETURNING TO INSTS; SAW ALT WAS 6300 FT WITH APPROX 600 FT RATE OF CLB. AURAL ALT WARNING WAS THEN HEARD. SAFELY BUT EXPEDITIOUSLY PUSHED NOSE OVER WITHOUT CAUSING FLT ATTENDANTS TO HIT CABIN CEILING. ACFT REACHED APPROX 6450 BEFORE ACTUALLY DSNDING TO 6000 FT. PASSING THROUGH 6300 FT AND DSNDING ATC ADVISED OUR ALT CLRNC WAS 6000 FT. AT THIS TIME WE WERE GOING THROUGH 115 HDG AND WAS GIVEN A TURN BACK TO 100 DEG NEVER REACHING THE 140 DEG HDG THAT WE WERE PREVIOUSLY CLRED TO. AUTOPLT WAS OFF. I BELIEVE DEV WAS CAUSED AS A RESULT OF FLC BEING DISTR WITH TCASII ALERT WITHIN A COUPLE OF HUNDRED FT TO LEVEL OFF ALT. IT WAS NOTICED A CORRECTION STARTED SECONDS BEFORE RECEIVING THE ACFT ALT AURAL WARNING BUT CLBING VECTOR RESULTED IN AN APPROX 450 FT HIGH DEV. SUPPLEMENTAL INFO FROM ACN 201594. THIS INCIDENT HAPPENED DURING A VERY BUSY DEP AS WELL AS ARR RUSH INTO ORD. ACR X; AN MLG; HAD DEPARTED RWY 32L ON A 040 HDG AND WAS LEVEL AT 5000 FT. APPROX 10 MI NE OF ORD; ACR X WAS CLRED TO 6000 FT AND TURNED R TO HDG 090. A FEW SECONDS LATER; A FELLOW CTLR OBSERVED ACR X CLBING THROUGH 6400 FT. THERE WAS ARR TFC ABOVE AT 8000 FT AND OFF HIS L AT 10000 FT. I IMMEDIATELY ISSUED INSTRUCTIONS FOR ACR X TO LEVEL AT 7000 FT AND ISSUED THE 8000 FT TFC. ACR X EVIDENTLY REALIZED HIS MISTAKE AS I ISSUED THE INSTRUCTIONS TO LEVEL AT 7000 FT AND HE DSNDED FROM 6600 FT BACK TO 6000 FT IN LESS THEN 2 SWEEPS (8 SECONDS). IT WAS AT THIS TIME HE INDICATED TO ME THAT HE HAD A TCASII ALERT. UPON FURTHER DISCUSSION WITH THE PLT; HE INDICATED HIS TCASII WAS GIVING HIM SEVERAL ALERTS SO BOTH PLTS WENT OUTSIDE THE COCKPIT FOR VISUAL VERIFICATION AND THEY FORGOT TO LEVEL AT 6000 FT. IN OUR ENVIRONMENT; 7000 FT BELONGS TO ARR AND FORTUNATELY HE HAD NOT DSNDED HIS ARR FROM 8000 TO 7000 FT. I FEEL THAT A TCASII DISTR CAUSED THIS PLTDEV FROM THE MOST BASIC OF PILOTING DUTIES; 'FLY THE ACFT FIRST.' THIS IS ANOTHER UNFORTUNATE OCCURRENCE WHERE TCASII HAS BECOME MORE OF A DISTR AND HINDRANCE IN A HIGH DENSITY TERMINAL AREA.