Business jet Captain reported receiving a terrain alert departing VNY when he was distracted by an ATC traffic alert call and failed to climb as per procedure.

Narrative

Took off from Runway 16R in VNY on the HARYS4 SID. There is a restriction to cross 1.5 DME south of the VNY VOR at or below 1750 ft. Altitude selector was initially set at 1700 ft. and a fix reference was input in the FMS so that we can climb to 4000 ft. once we reached the 1.5 DME.I leveled off at 1700 ft.; but while I was cross checking between the PFD and outside; I never caught the fact that we crossed the 1.5 DME and the altitude alerter was not set to 4000 ft. Of course ATC was talking to us at the same time and also we were alerted to traffic nearby. Because I was distracted momentarily; I missed the turn over PPRRY to heading 110 and continued straight. Seconds later; we received a terrain caution and ATC gave us an immediate turn to the left to avoid traffic. I applied climb thrust and initiated a climb and also a turn because of the TA. Once things stabilized in the climb; ATC had us copy a phone number and recorded it as a pilot deviation.This particular departure out of VNY can become overwhelming if not properly prepared for it. Requires a low altitude level off and then a climb a very short distance from the departure end of the runway. I have performed this departure many times in the past and never experienced any issues. For some reason this time; the way the chain of events unfolded caused an error. It's very important to brief this thoroughly on the ground and ensure both pilots are aware of what their actions will be.