Red or Brown would look perfect on you. Don't dye them blonde because it can draw the attention away from your eyes, while red or brown will enhance your eyes (red is complementary to green and the contrast will look amazing). As for black, I don't know. I need to see a picture of you to tell if it can really look good on you.
Red or blonde. Red will take away from your eye color (because red hair is on its own very eye catching), while blonde hair will make your eyes pop. So choose which one of these would you rather want to achieve. =)
I also have green eyes and fair skin and I had natural mousy brown hair but I dyed mine medium/dark brown with blonde ends (my photo) and it looked a lot better! Although, I think red would also look really nice, I was tempted to go red too. Please don't go blonde though!