ACR MLG FLC DEV DISTR FROM FLYING ACFT AND OVERSHOT THEIR ASSIGNED ALT. CREW WAS CONCERNED WITH TSTMS NEARBY AND FAILED TO 'FLY THE ACFT FIRST.'

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OUR CLRNC IN PART WAS TO FLY THE FLL 7 DEP AND MAINTAIN 3000 FT. JUST AFTER TKOF FROM RWY 9L FLL TWR TOLD US TO CONTACT MIA DEP; WHICH IN TURN TOLD US TO TURN L TO A HDG OF 360 DEGS. AS I WAS TURNING L I WAS WATCHING THE RADAR AND LOOKING OUTSIDE FOR HOLES TO PENETRATE THE AREA OF TSTMS ALONG THE EAST COAST OF FLORIDA. THE CAPT WAS ALSO WATCHING THE RADAR AND LOOKING FOR TFC THAT MIA DEP CTL TOLD US WAS AT 4000 FT AND I BELIEVE AT OUR 10 O'CLOCK POS. WHILE DIVERTING MY ATTN OUT OF THE ACFT FOR WX AND TFC; I INADVERTENTLY CLBED TO 3500 FT BEFORE RECOGNIZING THAT I CLBED THROUGH OUR ASSIGNED ALT. I THEN IMMEDIATELY DSNDED BACK TO 3000 FT. MIA DEP TOLD US TO MAINTAIN 3000 FT. THE ALT ALERT DID NOT SOUND OFF AS IT IS DESIGNED TO DO WHEN ASCENDING OR DSNDING ABOVE OR BELOW THE SELECTED ALT BY APPROX 300 FT. THE CAPT AND I DISCUSSED THIS SIT AND WE CAME TO THE CONCLUSION THAT ONLY HE SHOULD HAVE HAD HIS EYES OUT OF THE AIRPLANE LOOKING FOR TFC AND WX WHILE I (THE PF) SHOULD HAVE BEEN WATCHING WHAT WAS GOING ON INSIDE THE PLANE -- ALT; HDG; TCASII RADAR; ETC.