Captain reported an airborne conflict with a general aviation aircraft that departed the airport ahead of the reporters aircraft.

Narrative

Early morning departure from ZZZ to ZZZ1. First Officer (FO) leg to fly with Captain as pilot monitoring (PM). Standard briefing and all check lists were completed as per Standard Operating Procedures (SOP). FO was first flight as pilot flying (PF) in a Aircraft X since his IOE. Tower was closed and we called ZZZ Departure for a clearance once holding short of Runway XX #1 for departure at ZZZ. Departure gave us a release time of 5 minutes out. FO made the announcement on CTAF that we had to hold short on Taxiway X for 5 minutes. Aircraft Y behind us took Runway XX at intersection Y for a VFR departure to the northeast. As the Aircraft Y was taking off; departure cleared us for immediate release with release void if not airborne in 4 minutes. We were cleared direct to ZZZ2 3;000ft. We ran the before takeoff check list and lined up on Runway XX waiting approximately 3 minutes before starting take off roll. Runway XX has an obstacle departure procedure (ODP) to climb to 1;400 ft. MSL before turning south. On takeoff roll; the FO had forgot to press the TOGA switch and had no flight director guidance once airborne. I pressed the TOGA for him to bring up the flight director; and monitored airspeed. Passing through 1;000 ft. AGL; FO called for climb sequence and accelerated to 200 kts. We were passing through 1;400 ft. MSL; so I sequenced the FMS for direct ZZZ2 per the ODP armed NAV and started to retract flaps. FO started the turn towards ZZZ2...by this time; we were passing through 2;300 ft MSL. I noticed the traffic icon on the Multi-function Flight Display (MFD) when we received a Traffic Advisory (TA) as we started to turn. At the same time ATC commanded an immediate left turn and level off at 2;500 ft. MSL. We were already in a left turn and increased our rate of turn. I replied that we were passing through 2;800 ft. MSL at that time; and ATC told us to continue climb to 10;000 ft. Looking at track logs from Flight Aware; we executed our turn 1 NM behind and 200 ft. above Aircraft Y.The main cause of this event was an unexpected threat caused by the GA aircraft taking off on the same runway unexpectedly in front of us. This threat had not been briefed. A contributing factor was the pressure to take off within the narrow release window issued by departure. As PM; I was distracted when the flight director was not activated and focused on correcting/activating the FD; thereby delaying my sequencing of the FMS to provide lateral guidance to ZZZ2 for the FO and possibly a late turn out. This also distracted me from maintaining situational awareness and traffic avoidance; which contributed to us coming into close proximity of the GA traffic. Neither the FO nor myself anticipated the closure rate that we would have on the GA traffic. I should have initiated the turn in heading mode prior to sequencing the FMS; waiting for confirmation and then arming NAV and waiting for the FMS to start giving lateral guidance.Include a specified time frame in the SOP between any GA traffic departing prior to a scheduled departure from and uncontrolled airport. This could be a mandatory wait time irrespective of release time. Another suggestion would be to have scheduled flights depart only once Tower has opened; or if early morning flights are required; arrangements should be made with the city to have the Tower open at the scheduled times. From a personal perspective; any time an unanticipated threat such as this occurs prior to departure; I will require a full re-brief to mitigate the threat; even if it involves delaying the departure or waiting for a new release time.