I made the body of the cloak from a coat-weight, coarse-woven bit of wool. I'm not sure what kind of animal (sheep, goat, alpaca?) gave the fleece for the wool, but it smolders and goes out almost instantly when subjected to a burn test. It's wonderfully warm.
The fur around the shoulders is wolf. Yes, it's real. But it's also vintage, so if you feel bad about an animal dying to make this fur, then you can rest assured that it would long ago have died of natural causes by this time anyhow.
As you can see in this inspiration picture, the cloak is fastened by way of two leather straps which attach at the shoulders. I did a little research and believe (though I've not been shown any evidence from the GoT series or costumers) that the straps continue around the body and are fastened at the back with a buckle. I used two matching leather belts and they worked perfectly, though there weren't quite as wide.
It's a good thing I made my plan to construct this cloak way back in August, because it took me a long time to find all the pieces I needed -- especially the fur. That was a real steal at Goodwill for only $20.