With Länstrafiken in Norrbotten you can go by bus between Kiruna and Nikkaluokta. You find the time tables at www.ltnbd.se.
From Nikkaluokta it is a 19 kilometre long walk to the Kebnekaise Mountain Station. It is possible to go by boat (find the timetable here) a part of the way during summer and by snow-mobile or tracked vehicle during winter.
In summertime you can sign up at the mountain station for a guided tour to the summit. The tour takes about 8-10 hours and you need to be in good shape and at least 16 years old. More information at www.stfkebnekaise.com.