HS 125 Series
Flight Crew Multiengine; Flight Crew Air Transport Pilot (ATP); Flight Crew Flight Instructor; Flight Crew Instrument
Flight Crew Last 90 Days 80; Flight Crew Total 3900; Flight Crew Type 1259
Situational Awareness
ATC Issue All Types; Conflict NMAC; Deviation - Altitude Excursion From Assigned Altitude
Flight Crew FLC complied w / Automation / Advisory; Flight Crew Took Evasive Action
Procedure
Prior to departing; my F.O. (First Officer) called MKE clearance and got our IFR departure clearance to climb to 3;000 and upon entering controlled airspace; turn right heading 250 deg. We departed as instructed and after making our last call on CTAF; called up MKE Departure shortly after climbing through 2;000 for 3;000.  MKE confirmed radar identification and assigned us right turn 270.A few seconds after rolling out on that heading; I noticed a TCAS target at my 12 o'clock 3-5 miles and closing in 500 feet above my altitude. I was just beginning my level off with a few hundred feet left to my assigned 3000 feet. I quickly advised ATC of the target. He assigned me 'climb and maintain 6;000' with no mention to the aircraft I had just told him about. Then I received a TA from the TCAS. I advised him that I now have traffic in sight and I am descending. Then a few seconds later my TCAS gave me an RA to 'descend; descend now' (which I was already doing based on what I saw out my window prior to the RA). We ended up passing roughly 500 or so feet directly below an also westbound single engine piston (looked like a Cessna 172). When time permitted; I advised ATC that we got an RA on that plane.Had we had followed ATC instructions and climbed to 6;000 feet; especially if in some form of IMC; we would have almost certainly collided or come extremely close to it. The Controller did not seem concerned about the RA. I advised we were clear of conflict now and he simply told me to climb again to 6000 feet. No query of the RA.
HS125 Captain reported being alerted to an NMAC by a TCAS RA after shortly after departing the airport.