Kaçkar Mountains National Park is in the Rize province of the Black Sea Region of Turkey, bordering Georgia. Rize is the nearest city to the national park with an airport (RZV) and daily nonstop flights from Istanbul and other Turkish cities (see image below).
Award booking flights to Istanbul & Rize
Aeroplan miles can be used to book Turkish Airlines business class to fly to Istanbul and connect to Rize for 70K miles and low taxes.
Turkish Miles & Smiles can also be used to book the same flight, but with higher taxes (see image below).
LifeMiles does not support the Rize airport code on their website.
Rental Cars from RZV airport
Kayak has several rental car options available to book from RZV airport (link) for reasonable prices.
Best villages to stay and explore Kaçkar Mountains
I recommend staying at either Çamlıhemşin or Ayder, which are 20-30 miles from the Rize airport. Both places have good cafes, restaurants and nice boutique hotels that are reasonably priced at Booking.com (link) (link).
Ayder Plateau & Çamlıhemşin Scenery
Conclusion
Kaçkar Mountains National Park must be visited to explore its unspoiled nature, along with terrific dining and boutique accommodation options for a reasonable price.