B300 Captain reported loss of visual contact with previous departure resulted in a NMAC.

Narrative

While taxing to runway XX for departure; we observed a single engine Cessna depart runway XX. Upon reaching the departure end of the runway; we completed final checks and announced on CTAF that we were taking off on runway XX and would be departing the area to the east. The aircraft that had just departed advised that he was departing the area on upwind runway XX. My first officer had the other aircraft in sight at this point. We departed on runway XX. While climbing out; we began a left turn. At this point; we had lost visual contact with the other aircraft. As I continued to climb; in a left turn; I saw the other aircraft below us and off to the left. About the same time I saw the aircraft; I got a TA. I pitched up and continued to climb. We passed over the top of the other aircraft by an estimated 600 feet. I don't believe the other aircraft saw us.This incident illustrates the importance of the principle to see and avoid other aircraft. We were distracted during the takeoff and climb out and lost sight of the aircraft that had departed prior to us. I also believed that the other aircraft was further away from the airport and further southeast than what it actually was. Another contributing factor was the speed difference between the two aircraft; we were much faster. This incident was a good reminder for me to remain vigilant when operating in the vicinity of airports and especially uncontrolled airports; practice see and avoid and make accurate position reports. It is also a reminder to consider the speed and climb rate of other aircraft departing ahead of you.