Eurowings flies from Cologne / Bonn and Hamburg on scheduled flights to Klagenfurt. You can book flights and view current flight schedules at www.eurowings.com.
Easyjet will also bring you in winter 17/18 from London to Klagenfurt!
Other airlines have many flights to Graz or Ljubljana (Ryanair, EasyJet, etc.). You can also book inexpensive flights via Salzburg Airport (for example with TransAsia, TUIfly, etc.).
Transfer from the airport to your accommodation in the Klopeiner See - Südkärnten region is a breeze: the Carinthia Transfer takes you swiftly and smoothly from airport to your accommodation and back again.
Find your flight in the Klopeiner See - Südkärnten region: