If you want to try this, I would recommend trying on a very small area first. People HAVE reported skin irritation from topically applied Lugol's and the skin around your eyes is very thin and delicate.
Personally, I would not. I would look into other potential reasons for dark circles. The fact is, with your fair complexion, I would imagine that "dark circles" are somewhat normal for you. If the circles are darker than normal, I would look into why the blood is possibly pooling there, maybe you have circulatory problems. Use the CZ search box and search "dark circles"...
Liver problems, perhaps?