This is my very first critique, however, I have read a few and hope that I will be of sufficient help to the artist.
I like this picture. Not just because I like the character - being one of my favourites in Game of Thrones - but because the artist is gifted. The picture is done with photoshop, yet is looks just as if it was done with a brush and oil colours on canvas. That is the first thing I noticed about the picture.
Second is, there is emotion in it. Tyrion has an air of sadness about him in the picture, you can see it in his beautiful dark blue eyes, if you look close enough. This has been captured by the artist very neatly. The face of Tyrion Lannister is not a perfectly pretty one, like the one of his brother. Yet it is full of character: edges, beautifully shaped eyebrows, a soft and sweet mouth. All in all, I think, the face looks exactly like the one of Peter Dinklage does, which is not easy to succeed in.
The background is black and without any further ado, which is okay, because I think ornaments would have given the picture a less melancholic imagery, which would have been fatal. That way, the sadness in Tyrion's eyes comes across even better. The outfit of the character is unspectacular, which also avoids distracting the viewer.
What I like best about the picture is the hair. It has light strands in it, which seem to capture sun light that is unseen in the image itself. This gives some hope to the rather dark and sad picture.
All in all I think, the character was captured very neatly. All features are exact and pretty, background and clothes remain behind, which is good, because the viewer can concentrate entirely on Tyrion's expression.
There is a lot of sadness in the picture, but also a faint shimmer of hope.
Personally I object to the beautification of Tyrion. Was already surprised when I first saw him in the series... didn't fit the book's image too well. We should do the character the courtesy to keep him "ugly". He is all about overshadowing that with his wits, so I don't like to see an almost beautiful portrayed version. Of course this doesn't concern your skills as an artist, merely considering the subject in this case.