General aviation pilot reported a NMAC during departure with a helicopter that overtook and overflew them; requiring evasive action by the pilot.

Narrative

Departed ZZZ airport northwest-bound toward ZZZ1 airport. My intended route was to fly a 320 heading for a few miles and then turn westbound toward ZZZ1. During my takeoff roll I heard ZZZ tower clear a state trooper helicopter to depart from a taxiway northwest-bound. I made note that this was the same direction as my route. At approximately 5nm from ZZZ2; I was cleared to change frequencies. A few moments later I wanted to turn westbound toward ZZZ1; but I noticed on my TCAS that there was traffic in close proximity at the same altitude. I looked over my shoulder; but did not see any traffic. The helicopter was most likely obscured by the wing. I began a turn toward the west; but immediately got a traffic alert and stopped the turn. I could now see the helicopter and I turned to a heading which roughly paralleled their track. It appeared that the helicopter was attempting to overtake me. Soon after; the helicopter climbed above me. Once they were greater than 500ft above me according to TCAS; I resumed my turn toward the west. The flight concluded without incident. It would have been prudent for ZZZ Tower to advise one or both of us of the potential conflict. Particularly the helicopter as they were the faster of the two and likely to overtake.