Timetables for bus lines can be found in http://www.matkahuolto.info/lippu/en/
You can find companies in the business register of Kittilä that offer chartered transportation.
Meän Pirssi service line operates in Kittilä.
The nearest railway station is in Kolari, about 80 km away from Kittilä village. There are bus connections to Kittilä from the Rovaniemi railway station (about 150 km).
Timetables can be found in VR internet pages: