Kilpisjärvi is a village in Finland. It is a small village just before the Norwegian border, with quite a lot of traffic going to Norway. There is a shop, where many Norwegians come to buy cheap meat, alcohol and tobacco, so hitching to Norway is quite easy because of regular grocery trips. The shop is easy to locate, because it is the only one around.
North towards Tromsø
Hitching from the petrol station next to the shop can be succesful. If you can, try to get a ride to the border station a few kilometers away from the village. There is a lot of room to stop and even trucks might stop there.
South towards Muonio
Hitching towards south is really hard. There is usually no reason for the locals or almost anybody else to drive longer distances towards south, because Tromsø on the Norwegian side of the border is the biggest municipality around. The next bigger city on Finland's side is Rovaniemi over 400km away. Kilpisjärvi is a popular hiking destination, but usually hikers have their cars full of stuff and people. The locals have seen hitchhikers being stuck in the village even for a few days.
Definitely consider taking a bus at least to Muonio. Be early with the bus, it rides only once or twice per day and on some days it might not come at all.
Other useful info
There is a lean-to with firewood next to the Tundrea holiday resort. It is a good place to cook or warm up.