A HS125-800 PILOT ON THE TEB 5 SID WAS DISTRACTED BY HIS FO'S SQUAWK ERROR AND WITHOUT BACKUP BEGAN CLB EARLY TO 2000 FT. ATC CLEARED FLT TO 10000 FT.

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THE DEPARTURE OUT OF TEB IS CHALLENGING AT BEST; ESPECIALLY WITH WEATHER DOWN TO MINIMUMS. THE DEPARTURE WAS BRIEFED BY BOTH PILOTS MORE THAN ONCE TO ENSURE COMPLIANCE AND STAY OUT OF EWR'S AIRSPACE. UPON ROTATION AND SUBSEQUENT ENTERING OF IMC WITH PIC FLYING; ATC INFORMED US THAT A WRONG SQUAWK CODE HAD BEEN ENTERED. THIS COMPLETELY FRAZZLED THE SIC WITH CURSING; SLAMMING OF FLIGHT CAN ETC. TO LOCATE PROPER CODE FROM CLEARANCE. PIC WAS DISTRACTED BY THE COMMOTION IN THE RIGHT SEAT AND UPON REACHING 1500 FT ASKED SIC FOR DEPARTURE INSTRUCTIONS. IS IT 280 DEGS (HEADING) AND 2000 FT (ALTITUDE) WHILE NOT THE BEST PHRASED QUESTION AS A PILOT IT THE MESSAGE WAS QUITE CLEAR WITH THE TEB 5 DEPARTURE INSTRUCTIONS; ALTHOUGH INCORRECT BY FORGETTING THE 4.5 DME RESTRICTION. SIC DID NOT LOOK AT THE DEPARTURE AND SAID YES; FLY IT WITH THE TURN TO 280 CORRECT THE ALTITUDE WAS NOT TO BE CLIMBED TO UNTIL 4.5 DME FROM THE TEB VOR. THIS EARLY ASSENT LED TO ATC INSTRUCTING US TO CLIMB TO 10000 FT THRU EWR'S AIRSPACE EARLY TO AVOID INCOMING TRAFFIC. NO FURTHER INCIDENTS HAPPENED ON FLIGHT.