Air carrier pilot reported an NMAC shortly after takeoff from JFK when a helicopter passed 200 feet below them.

Narrative

Prior to being cleared for takeoff from JFK Runway 31L; JFK Tower advised us that there would be a helicopter flying low level that would remain north of the runway. After takeoff while flying the SKORR5.YNKEE departure at approximately 1;000 feet; our TCAS showed a proximate threat of air traffic 200 feet below us almost directly under our position; very slightly to our right. That target threat then turned into a TCAS TA. We were climbing through 1;100 feet at this point and beginning our left turn in accordance with the SKORR5 SID. The TA went away after a few seconds and we continued the departure without further incident.Due to the nose-high attitude for initial climbout and subsequently turning left on the SID; we never had the opportunity to acquire the traffic visually.As we checked in with NY Departure; we advised them that we passed over the helicopter by about 200 feet and Departure said they would relay that to JFK Tower.Once above 10;000 feet and after our workload decreased; I advised Dispatch via ACARS that we had what I considered a near midair collision with a helicopter. Dispatch acknowledged and advised that there was a helicopter at that position during our departure.Later we replayed the flight path of the helicopter and it appeared that; at the time of the incident; the helicopter had possibly touched down in the GA area off the north side of 31L departure end (showed altitude of 0 ft); then climbed out underneath of us. I suspect that the helicopter failed to stay north of the runway and caused the loss of separation.