Here are some suggestions you can try for getting hair dye off of your skin -- just be sure to avoid using any products that are too harsh on your face.
1. Use liquid dish soap or laundry detergent. This method is best suited for use on the hands since detergents can be a bit too strong for the delicate skin of the face. Simply rub the detergent into the stains until they start to lift, then rinse with clear water to wash the detergent away. 
2. Try using baby oil or olive oil. The oil will help break down the dye so that it can be washed away. Simply apply the oil to a washcloth and use it to gently rub away the stain. Once the stain has lifted, use soap and water to wash away the remaining oil. 
3. Use nail polish remover for stains on your hands. The acetone will quickly break down the dye and remove the stain. Just avoid using nail polish remover on your face since the skin there is quite sensitive. 
4. Rub the stains away with petroleum jelly. Again, just apply a thin layer of petroleum jelly to the stains and rub them gently with a washcloth or cotton ball until they fade away.
5. Exfoliate the stains away with toothpaste. Try using a toothpaste with baking soda in it to gently exfoliate away the stain. Just apply a small amount to your skin and rub it over the area. As the dead skin cells are buffed away, the dye will begin to fade. You can use this technique on either your face or hands. 
6. Try hydrogen peroxide. The bleaching power of hydrogen peroxide can help lighten the dye, removing the stain from your skin. The dermatologist in this video does a great job of explaining how it works:
Good luck getting the dye off!