Light Jet Captain reported a NMAC with multiple aircraft during departure that required evasive action.

Narrative

I was on an IFR flight plan and cleared the departure from ZZZ. The SID directs climb on runway heading to 500; then a right turn to heading 220 and climb to 3000. I heard Tower talking to a couple of VFR aircraft arriving from the Southwest. When Tower mentioned to the one that I would be a conflict; the pilots replied 'Well; I'll start my descent then.' I was subsequently cleared for takeoff. As I turned right in accordance with the SID instructions; I immediately received two traffic alerts on my TCAS. The first aircraft was right off my nose; less than 2 NM; and slightly above my altitude as I was climbing out at approximately 3;000 ft/min (per normal procedures). I entered a steep right bank (approximately 50 deg) and pitched up to 15 deg nose high to avoid. I rolled out on a 250 heading to avoid a second aircraft at approximately 2;500'; now immediately to my forward right with a high rate of closer. I never visually acquired either aircraft due to my evasive maneuvers and the rapid closure rate. The first target came close enough that the 'alert' diamond merged with my aircraft symbol on my display in a 5 NM scale. The second aircraft was approximately 1 NM to my right as I climbed to 3;000 and quickly departed the area. I notified both Tower and Approach that I was 'deviating from the SID for traffic.' Nothing else was mentioned over the radio. I did call the ZZZ Tower after my landing and spoke with the supervisor on-shift. He thanked me for the feedback.