Public transport:
The most sustainable way to get there is by train to Oberdrauburg (coming from Lienz or Villach) or to Hermagor (coming from Villach).
From Oberdrauburg or Hermagor, you can then reach Kirchbach by bus.
Details at - Nassfeld mobility information as well as Kärntner Linien route planner
Arrival by car:
From Hermagor via the B111 (Gailtal Road) or from Oberdrauburg via the B110 (Plöckenpass Road) to Kötschach-Mauthen and from there further via the B111 (Gailtal Road) to Kirchbach.