Greenland has only one fresh water species for flyfishing: Arctic char.
The char has an annual migration from fresh to salt water and back again. They enter the salt water early in the season, stay in salt water over the summer and start migrating back up to the fresh water in mid-July.
Some of my favourite flies are:
Purple Egg sucking leach tied dog nobler style. This is an Alaska fly that works well for a lot of different fish, including char. I tie it on a size 8 dog nobler hook, I guess the dog nobbler hook is no longer being produced but any long shank hook is fine. The heavy eyes makes it behave similar to a jig.
I have my own version of the egg sucking leach that i call the nobody. It is simply an eggfly with heavy eyes and a marabou tail.
For smaller rivers I like to fish the old favorite Teal Red and Silver. It is a bit more old fashioned and a bit more cumbersome to tie than some other flies that I like to tie. But it works. Another classic in the same style the Alexandra works just as well.
Surface fishing is an option some places. The old standby Elk hair Caddis works very well. I like to fish it streaking.
Foam flies work just as well as caddis flies in most situations.