A PVT JET SMT OVERSHOT ITS ALT ON CLBOUT.

Narrative

I HAD JUST RECENTLY BEEN QUALIFIED AS A COPLT IN THE SMT JET. I WAS FLYING THE ACFT OUT OF SAF AND WE WERE CLRED TO 11000 FT. I WAS MAINTAINING A SCAN INSIDE THE COCKPIT; ADJUSTING PWR; MAINTAINING PROPER CLB AND THOUGHT ALL WAS IN ORDER. MY ATTN WENT MOMENTARILY OUTSIDE THE COCKPIT LOOKING FOR OTHER ACFT WHEN WE BOTH SAW THE TFC AT 1-2 O'CLOCK; APPROX 2 MI; ABOVE US; IN THE OPPOSITE DIRECTION. I IMMEDIATELY CHKED THE INSTS AND REALIZED WE HAD CLBED TO 13000 FT. WE WERE SURPRISED WE HADN'T HEARD FROM CTR NOR DID THE ALT ALERT GO OFF. WE PROMPTLY NOTIFIED CTR OF OUR ALTDEV AS WE DSNDED TO OUR PROPER ALT. THIS AIRPLANE IS THE ONLY JET I HAD FLOWN AND I WAS NOT YET ACCUSTOMED TO THE RAPID RATE OF CLB. I ALSO BELIEVE I NEED TO MAINTAIN A BETTER SCAN OF THE FLT INSTS. IN THIS INSTANCE; I WAS EXPECTING THE ALT ALERT TO CATCH MY ATTN IN THE EVENT I DID NOT SCAN PROPERLY. UNFORTUNATELY; THE ALT ALERT DID NOT WORK. SINCE THIS INCIDENT; WE HAVE DISCUSSED OUR COCKPIT PROCS ON ALT CALLOUTS TO INSURE THAT THIS NOT HAPPEN AGAIN.