Lucas Girardet recommends Kangerlussuaq – Sisimiut, 180km (112mi), as one of the best hikes in the world.
This route is the classic hike in Greenland for the hardened wanderer and others who like challenges.
The route goes through landscapes which vary a lot. From the dry inland at Kangerlussuaq to the rolling ground with the high hills by the coast. Considering the length of the hike, approx. 7 – 10 days, it is a hard hike which should not be challenged unless you have made preparations from home.
Kangerlussuaq is the settlement with the biggest and best airport in Greenland. It’s the usual gateway for those flying into Greenland. From there, you could fly one way to Sisimiut, and hike back.
Sisimiut is a port town known for mountains towering up from the sea, fjords and glaciers.
The hike sounds awesome though not inexpensive. Of course the hiking season is quite short, as well.
filipholata from the Czech Republic posted some photos of Greenland on flickr:
besthike.com 
This is beautiful. I wish I could take my family there one day.