OVERSHOOT OF THE INITIAL ALT ON THE TEB5 DEP.

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AFTER TAKING OFF FROM RWY 24 AT TEB ON THE TEB5 DEP; WE MISSED OUR INITIAL ALT TO LEVEL OFF AT 1500 FT AND CONTINUED OUR CLIMB TO 2000 FT. WE WERE ADVISED OF OUR MISTAKE BY NY DEP. NO OTHER ACTION WAS PROMPTED BY OUR MISTAKE AND WE WERE CLEARED TO MAINTAIN 2000 FT AT THAT TIME. THIS WAS A LIFE GUARD FLT AND TIME WAS OF A CRITICAL NATURE. THE MEDICAL TEAM ARRIVED 45 MINUTES EARLIER THAN SCHEDULE WHICH CREATED A HURRY UP ATMOSPHERE. EVEN THOUGH I; THE PF; BRIEFED THE DEP WITH MY COPLT IN THE RUSH OF THINGS WE GOT CAUGHT UP AND MISSED THE LEVEL OFF AT THE REQUIRED ALT. WE HAVE DISCUSSED THE MATTER WITH MY FO AND I REALIZE THAT IT DOESN'T MATTER HOW MANY TIMES WE HAVE FLOWN INTO THIS ARPT IN THE PAST (WHICH WE DO IN A REGULAR BASIS) WE SHOULD NOT LET OURSELVES GET CAUGHT IN THE RUSH OF THINGS; AND PAY MORE ATTENTION TO OUR DEP PROCEDURES.SUPPLEMENTAL INFO FROM ACN 738424: THE ALTITUDE PRESELECT WAS SET TO 2000 FT INSTEAD OF 1500 FT AND THE PF CLIMBED TO 2000 FT. IF I; AS FO; HAD CHOSEN 1500 FT IN THE ALTITUDE PRESELECT THE PF WOULD HAVE BEEN REMINDED TO LEVEL OFF UNTIL 4.5 DME. THAT WILL BE MY CORRECTIVE ACTION FROM NOW ON WHEN MY JOB TAKES ME TO TEB OR ANY OTHER AIRPORT WITH A SIMILAR DEPARTURE.