hmmm French Press at a campground? Been there done that. It's kind of messy to clean up.

First thing I do is heat a pot of water on the camp stove. I use my old Melita filter holder set over top an old fashioned aluminum camp style percolator (guts removed), and pour the boiling water into that. Once the coffee is made, swish some of the remaining boiling water into a thermos bottle to warm it, pour it back into the pot (save for dishwashing) then decant the coffee into the thermos bottle to keep it hot for as long as it's in there. A 2nd thermos is useful for hot water - you can use to wash your face, freshen up a little after bike ride, or wash the dishes.