AN ACR FLT CLBS ABOVE ITS ASSIGNED ALT WHILE THE CREW IS BUSY WITH TSTM AVOIDANCE N OF PHX; AZ.

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WITH TSTMS IN AREA; I DECIDED TO LET THE FO FLY THE LEG PHX-DEN. EVEN THOUGH FO WAS RELATIVELY INEXPERIENCED; WITH THE WX I WANTED TO RUN THE WX RADAR AND COMMUNICATE WITH ATC. AFTER TKOF IT BECAME EVIDENT THAT WE WOULD NEED TO DEVIATE TO MISS LARGE BUILDUPS WHICH WERE IN OUR FLT PATH. I ASKED FOR PERMISSION TO DEVIATE AND WAS GRANTED SAME. OTHER ACFT IN THE AREA WERE ALSO DEVIATING. ATC WAS VERY BUSY. WE WERE CLRED TO 15000 FT. WHILE CLBING; I WAS HEAD DOWN ADJUSTING THE WX RADAR AND TALKING TO ATC. ALL OF A SUDDEN; I HEARD THE ALT ALERT ANNOUNCING A DEV FROM ASSIGNED ALT. I LOOKED UP TO SEE US GOING THROUGH 15500 FT AND STILL CLBING AT ABOUT 2000 FPM. I IMMEDIATELY TOOK THE ACFT AND STARTED A DSCNT BACK TO 15000 FT BEFORE ASKING THE FO TO RESUME CTL. DURING THIS PROCESS; WE GOT UP TO APPROX 15800 FT. ATC REMINDED US OF OUR ASSIGNED ALT AND WE ACKNOWLEDGED. WE LEVELED AT 15000 FT AND I ASKED THE FO TO ENGAGE THE AUTOPLT. LESSONS: BE A LITTLE MORE AWARE OF FO EXPERIENCE LEVEL. CONSIDER HAVING PF USE THE AUTOPLT IN HVY WORKLOAD SITS. IF I COULD DO IT OVER; I WOULD HAVE ENGAGED THE AUTOPLT AFTER TKOF. SUPPLEMENTAL INFO FROM ACN 448186: I WAS HAND FLYING AT THE TIME AND PREOCCUPIED WITH TSTM AVOIDANCE. LESSON LEARNED WAS TO UTILIZE THE AUTOPLT TO LESSEN WORKLOAD WHEN TASKED/DISTR WITH FACTORS SUCH AS WX AVOIDANCE.