Corporate Captain reported a NMAC while taking off from a non towered airport. The Captain stated becoming aware to late; of the twin engine aircraft on short final; and started the takeoff roll. Then during the initial climb the other aircraft was directly above the Captain's jet; as they were executing a missed approach straight ahead.

Narrative

I was flying with a high time; (but low time in the aircraft) contract pilot. This was our 6th leg of the day; departing ZZZ Runway XX empty. I was in left seat as pilot in command (PIC).On our initial call to ZZZ clearance in the chocks; we were given an expected departure window of XB03 to XB06. We taxied early and got to the hold short about XA55. When we checked in at the hood short - ZZZ clearance said 'If you can get airborne in the next 30 to 60 seconds you are clear for an immediate takeoff; Aircraft Y on a 4 mile final. My second in command (SIC) answered 'Roger; insight'I cleared the left side of the aircraft; asked my SIC to make a Unicom call; and I completed the takeoff checklist. As I was starting the 90 degree turn onto the runway - I looked to my right and saw that the Aircraft Y was much closer than 4 miles - more like 1 mile! I immediately applied takeoff power in an attempt to get out of his way. I thought he possibly still had separation to land if we immediately took off.But the Aircraft Y broke off the approach. Instead of turning crosswind - the Aircraft Y went straight ahead. As we were crossing the departure end of the runway after getting airborne; I was looking left expecting to see the Aircraft Y crosswind or even downwind. Instead; I looked directly overhead to see the Aircraft Y directly overhead. If I had continued climbing we would have hit this aircraft. We were probably 500 ft. away.A lot of mistakes in this event. The first is with me - 6th leg of the day; empty; on a rescue mission to pick up a company member in ZZZ1 who was stranded by a cancelled charter. 1) We were in a hurry; I was in a hurry. I didn't show it until I took the bait - cleared for takeoff if you can get airborne in the next 30 to 60 seconds!' 2) I took the runway before we make a clear UNICOM call 3) I did not clear the right side of the aircraft myself; taking my copilots response to the 'Aircraft Y on a 4 mile final' as a clearance of the right side of the aircraft.4) My contract copilot should never have been in our aircraft. He was legal. But he misrepresented his experience with our avionics suite. He was a hindrance all day.5) My contract copilot never said a word about the traffic. He rogered the call of 'Aircraft Y on a 4 mile final' with a call of 'insight'. I took that as a confirmation that the Aircraft Y was at 4 miles. It wasn't. It was closer to 1 mile; maybe 1.5 miles at the most.He should have said 'stop.' Before I taxied onto the runway.6) We were both monitoring UNICOM on the number 2 radio during our entire taxi. We never heard a single UNICOM call from this Aircraft Y. 7) When the Aircraft Y decided to break off the approach - they should have immediately turned crosswind. Instead they went straight ahead - didn't begin the crosswind turn until past the departure end of the runway. Additionally - they didn't climb much; they were well below pattern altitude. We were at about 200 ft. at the departure end when I saw them directly above us."