Light Transport; Low Wing; 2 Turbojet Eng
Flight Crew Multiengine; Flight Crew Instrument; Flight Crew Air Transport Pilot (ATP)
Flight Crew Last 90 Days 140; Flight Crew Total 13500; Flight Crew Type 750
Distraction; Situational Awareness; Communication Breakdown
ATC Issue All Types; Conflict Airborne Conflict; Deviation - Altitude Overshoot; Deviation / Discrepancy - Procedural Published Material / Policy; Deviation / Discrepancy - Procedural Clearance
Environment - Non Weather Related; Human Factors; Procedure
This flight started the night prior when my normal First Officer had flown someone a few days earlier who contracted COVID-19.  That night; I was left trying to find a new First Officer for the next morning's flight at XA:00am.  I found an FO who has not flown in a while.  Our flight to ZZZ was uneventful.  On climbout from ZZZ the Tower Controller had issued a speed of 230 kts and a climb to and maintain 10;000 feet.  I asked the FO to run the climb checklist. While doing so; the Controller issued us instructions.  My FO was caught off guard and looked at me as if to say; 'I didn't hear that.'  I thought I heard and understood the instructions to be 'Normal speed and climb 12;000 feet and I read back 'Normal speed; climb 12;000 Aircraft X'.  I then began to climb.  This put me into conflict with Aircraft Y that was in the downwind for ZZZ; I never got a TCAS or TA/RA alert from my TCAS system.I know that after the instruction to proceed with normal speed; there was another instruction after that statement.  I just cannot remember what it was.  The crew change and heavy workload contributed to the misunderstanding.  I do not know what could have been done differently to avoid this in the future.  I do wonder why ATC would have pilots turn left towards another aircraft even if they were planning to keep the aircraft separated by 2;000 feet. I have flown this departure from ZZZ many times in the past and the point where we were issued normal speed was completely normal from past experience.  I really wish that ATC would issue a climb straight ahead until the departing aircraft is no longer in a position to potentially interfere with other aircraft."
Corporate Captain reported an incorrect read back for altitude which caused their aircraft to climb an additional 2;000 feet resulting in an airborne conflict with another aircraft.