CL60 CREW DEPARTED TEB ON RWY 24 DIRECTLY INTO A TSTM CELL. THE CREW HAD AN ALT OVERSHOOT AFTER MISSETTING THE ALT ALERT SYS AND MISREADING THE TEB 5 SID.

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ON DEP; I BUSTED A 1500 FT MSL ALT RESTR; BRIEFLY LEVELING AT 2000 FT BEFORE IMMEDIATELY RETURNING TO 1500 FT. TKOF WAS ON A SID OF RWY 24 TOWARDS A THUNDERSHOWER DIRECTLY OFF THE END OF THE RWY. CLBED STEEPLY TO GET TO TURN ALT IN ORDER TO MAKE TURN AWAY FROM TSTM. (HAND FLYING ACFT.) IN THE TURN; OVERSHOT ALT BY 500 FT. I HAD MISREAD THE SID AND THOUGHT WE WERE SUPPOSED TO CLB FROM 1500-2000 FT AFTER PASSING THE 4.5 DME ARC. I HAD THEREFORE SET THE ALT ALERTER TO 2000 FT; PLANNING TO LEVEL OFF AT 1500 FT ON MY OWN UNTIL 4.5 NM. IN THE DISTR OF URGENT TURN; OVERSHOT ALT. ERROR CHAIN: 1) MISREADING A COMPLICATED SID. 2) SETTING THE HIGHER OF PRESUMED ALT RESTRS. 3) BECAUSE 2000 FT WAS SET IN THE ALT ALERTER (INCORRECTLY); NO FLT DIRECTOR; OR REMINDER WOULD PROMPT LEVELOFF AT 1500 FT. THIS IS A COMPLETE AUTOMATION PROB: IMPROPER PROC FOR PHASE (OR LACK OF USE) OF AUTOMATION SELECTED; IE; I SET MYSELF UP FOR AN ERROR WITHOUT REALIZING IT. IN THE FUTURE; I WILL READ THE SID CLOSER AND SET THE ALT ALERTER TO THE LOWER RESTR. YOU CAN ALWAYS MOVE IT TO THE HIGHER ALT AFTER 1ST LEVELOFF.