If you are arriving in Thailand via Bangkok easiest is to fly to Chiang Mai.
Please be advised there are two airports in Bangkok.
Most international flights arrive in Suvarnabhumi International Airport. There are a few air carriers who offer domestic flights to Chiang Mai from here.
Many domestic flights in Thailand are offered from Don Mueang International Airport.
Currently there is a shuttle bus between airports but factor in time/traffic if trying to connect. (And this is of course subject to change)
There are also direct flights from a few of the larger islands to Chiang Mai if you decide to arrive a few days early or stay a few extra days for some beach. (Highly recommended)
I would recommend a night or two in Bangkok or Chiang Mai to acclimatize and overcome your jet lag.
For a travelers who seek a bit more "adventure". There is also the possibility of travelling by train from Bangkok to Chiang Mail, daily and overnight trains are possible. Buses are also possible.