C525 CREW OFF RWY 24 AT TEB FLY TEB 5 SID INCORRECTLY AND RECEIVE A TCAS TA ON AN AIRLINER LNDG EWR.

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AFTER TKOF FROM TETERBORO; WE WERE ADVISED BY ATC TO TURN TO 290 DEGS. AT THE SAME TIME; I NOTICED ALT WAS OFF 400 FT. TRYING TO CORRECT THE ALT; I WAS SLOW IN THE TURN (ONLY STANDARD RATE). AS WE WERE IN THE TURN AND DSCNT; THE TCAS ALERT WENT OFF. CAUSES: #1) DISTRACTION ON THE FLT DECK -- RUNNING CHECKLISTS; TURB; HIGH VOLUME TFC AREA (LOOKING FOR OTHER ACFT); SENSITIVITY TO PAX'S NEEDS ON ALT CATCH; LENGTH OF DUTY DAY. #2) 5000 FT WHICH WAS PUT INTO ALT HOLD; HUMAN ERROR; MISREADING DEP PROC. SOLUTIONS: #1) FLY THE AIRPLANE FIRST -- DELAY USE OF CHECKLIST UNTIL LATER. #2) DOUBLE CHECK ASSIGNED ALTS. #3) WORRYING TOO MUCH ABOUT SMOOTHNESS -- GET ON ALT AND HDG. SUPPLEMENTAL INFO FROM ACN 606436: CLRNC GIVEN WAS TEB 5 (RWY HDG TO 1500 FT TURN R 280 DEGS). PF FLEW THROUGH ALT TO ABOUT 1900 FT; CREW RECOGNIZED ALT AND BEGAN CORRECTING; BUT FAILED TO INITIATE TURN IN TIMELY MANNER. ENTERED TURN STANDARD RATE AND WAS ADVISED 1500 FT 290 DEGS BY ATC. TCAS ALERT ACTIVATED; ACFT RATE OF TURN INCREASED; AND ATC ASKED IF WE WERE IN TURN. WE BELIEVE A CONGESTED AREA; AT NIGHT; AND FATIGUE TO BE CONTRIBUTING FACTORS. CREW BECAME DISTRACTED BY ALT DEV AND LED TO A HDG DEV. A PRE TKOF BRIEFING WAS PERFORMED; YET GIVEN THE CONGESTION OF THE AREA; A MORE THOROUGH OR REPEATING OF CERTAIN KEY ELEMENTS OF THE DEP WOULD HAVE HELPED KEEP THOSE ELEMENTS FRESH IN OUR MIND. THE CREW HAD RESTED IN 'DAY ROOMS;' YET NEEDED TO BE MORE MINDFUL OF OUTSIDE FACTORS: #1) LAST LEG OF DAY; #2) NIGHT; #3) CONGESTED AREA. EXTRA EFFORT/AWARENESS OF BEING IN THESE OUTSIDE FACTORS WAS NEEDED. CALLBACK CONVERSATION WITH RPTR ACN 606325 REVEALED THE FOLLOWING INFO: RPTR STATED THEY WERE A PART 91 OP ON A C525. AFTER TKOF; THEY RECEIVED A TCAS TA ON AN AIRLINER THAT APPEARED TO BE 2 MILES AWAY ON APCH INTO EWR AND WERE GIVEN A TURN BY ATC TO HDG 290 DEGS. RPTR SAID THEY HAD OVERSHOT TO 1900 FT AND BEGAN A SLOW DSCNT TO 1500 FEET. HE SAID THEY WERE ALSO TURNING A LITTLE SLOW FOR PAX COMFORT. RPTR SAID FOR SOME REASON THE OTHER PLT HAD SET THE ALT TO 5000 FT. RPTR SAID THEY WERE RUSHED; BUT SHOULD HAVE SPENT MORE TIME REVIEWING THE SID AND DONE A BETTER JOB WITH THE CHKLIST. WHILE ENRTE; THE PLT SAID ATC ASKED THAT THEY CALL AFTER THEY LAND. HE SAID THE OTHER PLT TALKED TO ATC AND WAS TOLD THEY WOULD GET BACK TO THEM. RPTR ADDED THAT THE DEPARTURE FREQ ALWAYS SEEMS TO BE CONGESTED. HE SUGGESTED A SECOND FREQ BE USED FOR FASTER COMS WITH ATC. THE RPTR ALSO SAID THE DEP CHART TEXT WOULD BE EASIER TO READ IF EACH RWY PROC WAS IN A SEPARATE PARAGRAPH.