EMB-145 Captain reported they climbed above their assigned altitude resulting in a TCAS/RA for a VFR overflight.

Narrative

We were operating Flight XXXX out of ZZZ. They were departing runway XXR with calm winds. We requested to depart runway XY in order to try to make up for being delayed from maintenance issues earlier that had caused a rejected takeoff in ZZZ1 with further delays for lack of staff afterwards. We were cleared for takeoff on runway XY (3253 ft elevation) and fly runway heading to 6000. We departed and were switched to departure at about 800ft agl. After switching and before we could call them; we immediately heard Departure calling us with their traffic alert going. This was highly unusual and safety was compromised. We responded quickly and at about 1000ft agl they advised immediate right turn to 130 for traffic. I was concerned because it seemed like they were turning us toward the Cessna traffic who was proceeding north from southeast of the field. We advised traffic not in sight and we began getting traffic alerts at about 4600 ft msl and we still heard the ATC traffic alert going off in the background as they spoke to us. I reduced our rate of climb and we attempted to located the traffic visually per the AOM. The Cessna had us in sight and was told to maintain visual separation. The Cessna said they would comply . ATC kept giving us traffic direction but did not change our heading. We figured ATC was taking us behind the traffic. Both our and ATC alerts kept going off as things were happening quickly and becoming a little confusing as the situation should have been getting better but it wasn't. Because of this; the FO kept trying to ascertain the location of the Cessna from ATC. The demeanor from ATC and the other aircraft suggested that we would clear but the alerts suggested otherwise and we were hyper focused on the rapidly deteriorating safety margin trying to visually acquire the Cessna in order to assure separation. We again were given a location for the Cessna and told it was at 6500. We did not see it for a few seconds. We then got an RA to descend. As we got the RA we were able to visually acquire the Cessna. I complied with the RA and we were clear of the contact quickly and level in about 300 ft at 6000ft. As we started the RA; we advised ATC the traffic was in sight and after the RA and they cleared us to 17;000. We then realized I had accidentally flown thru 6000 ft trying to locate the traffic visually by about 325 ft. This was inadvertent and not intentional and was an unfortunate result of trying to visually acquire and gain visual separation from the traffic in a confusion and rapidly deteriorating situation that should have been resolved initially by proper ATC vectoring and wider initial altitude separation. We did not hear the 1000 ft prior signal due to the traffic alert going off and the FO was rightfully questioning ATC on the Cessna. Without the signal during a high task saturated and threat scenario we missed the 1000ft call out. It should be noted that when I finally saw the Cessna; it looked to be about 2 miles at our 11:30 clock position on about a 330 heading. Our reciprocal heading would have been 310. For an aircraft that was on an original heading of north and given maintain visual separation; this heading does not appear as an appropriate response by the Cessna to maintain visual separation. It's the wrong direction to increase separation when he was initially southeast of the field. This may have been a reason for the confusion as to why the situation deteriorated from initial vectoring and perhaps ATC not realizing the avoidance vector was not working out sooner than later. I also believe that ATC launched us into an immediate bad situation where aircraft separation was compromised. Tower cleared us for takeoff about 1.5 minutes before our release time expired. ATC Tower may have felt operational time pressure to get us out(we had not rushed them). I believe a Swiss Cheese model developed from the moment takeoff clearance was given. I regret and accept that we accidentally went above our6000 ft altitude; but the many externally unusual and nonstandard factors in a rapidly deteriorating scenario with safety margins-reducing; bad vectoring; possible faulty maneuvering by the Cessna; minimal initial altitude clearances for both aircraft and being cleared for takeoff into poor aircraft separation are what created multiple threats and distractions all within 3000ft of the ground while trying to visually acquire the Cessna to assure separation are what lead to the missed level off.Suggestions: From this event; for future similar events and Traffic Alerts; I learned that I will use automation and engage the autopilot to help back up flying while trying to locate threats especially when ATC and aircraft systems are both giving traffic alerts. This will allow more focused attention overall on the threat and flight parameters especially during rapidly changing task saturated moments. Also; When the traffic is not in sight and the situation doesn't start to resolve in a satisfactory time frame; I will ask ATC for more aggressive vectors away from the target. I will also query distance to target . In addition if both ATC and aircraft systems are giving alerts; I will ask ATC to have the threat aircraft say heading. It's possible that during this time of government shutdown; ATC may have been short Staffed or had inexperienced personnel or both on duty which could have lead to us getting an inappropriate takeoff clearance. This is not known; but could be a factor at any airport. So while the shutdown is in effect or known short staffing is known; I will not ask for clearance for other than the duty runway when the winds are calm.