The real key is to empty the cup into the water so that the blood flows away from the camp. Wash with drinking water. Use hand sanitizer to wash your hands, but be sure to rinse well.
After the camping experience, I have a few more suggestions to add: Bring a nail brush and bio soap so you can clean your hands well before removing the cup. I couldnt' BELIEVE how long it took me to get my nails clean after that trip! I'd even suggest bringing gloves or finger cots to combat the dirt or hang-nail factor.
The cup is FAR superior to tampons and pads when camping. Tampons you have to stop and change too often, and you have to carry then with you until you can burn them in the campfire. Pads you have to triple baggie and pack out with you.
If you're camping with Aunt flo, bring your cup!