Hawker 4000 flight crew reported complying with a TCAS RA during departure climb out from a non-towered airport.

Narrative

At approximately XA00 hours we taxied from the main ramp to holding short of Runway XX at ZZZ. My First Officer (FO) and I had discussed with the surrounding airspace it would be best to depart IFR even though we were in VMC conditions. The First Officer picked up our clearance on frequency on the ground in ZZZ. The clearance was cleared to ZZZ1 via radar vectors to ZZZ2 then as filed climb and maintain 6000 expect FL 180 in 10 minutes contact ZZZ Approach; clearance void if not off in 5 minutes cleared for release. We then took Runway XX for departure. It should be noted a small Cessna had just departed prior to us and I saw them make a right crosswind departure. The initial take-off was uneventful until my First Officer contacted ZZZ. The Controller asked about our make and model aircraft; like we were pop up VFR traffic and did not seem to know who we were. At the same time we heard ZZZ trying to contact us on our other radio which was still tuned to that frequency. As my First Officer spoke to ZZZ I received a traffic alert while passing through 3000 ft. I immediately complied with the command and leveled off to avoid the conflict. After the conflict was over I continued my climb and started a left hand vector given from ATC up to 9000. The rest of this flight to ZZZ1 was uneventful. Two things happened here initially ATC provided us with the wrong frequency and I did not report the alert to ATC and had deviated slightly from my clearance to avoid the conflict.