This is a very good question, the answer is far from trivial, it is treated in great detail by W. Pauli in his excellent book "Theory of relativity". (pages 96-97). Get the book, it is well-worth the few dollars it costs.
For your case (simpler than the one treated in the book), your image will appear color shifted towards the blue spectrum. The reason is that light reflecting off you gets blue shifted when it hits the incoming mirror and it gets blue shifted again when it gets reflected from the mirror back into your eye. So, you get a double blue-shift effect:

If you do the experiment with the mirror receding from you, you will get a double red-shift effect:
If the relative speed between you and the mirror v, is a significant fraction of light speed,c, the effect is noticeable.
There are many cool websites that show simulations of the above.
This one is one of my favorites.