A scenic campground about 7 miles away and 2,600 feet up from the North Quinault Trailhead. There are several sites to choose from, most of which are large enough to accommodate two 2-person tents each. Since you're above 3,000 feet, no campfires permitted, and hammock campers have fewer options than they'd have at riverside campsites within ONP. No toilet to speak of -- at least none marked -- that I saw in the one night I stayed there. Campsites are spaced apart enough to give a good sense of privacy too. Good basecamp to explore Tshletshy Ridge or Finley Peak.
Lastly, while the you can use the lake for a water source, there's always the logistical challenge of getting to water deep enough to fill a bottle without getting too much sediment but not so deep that your shoes get filled with water. If you want to avoid this, you may have to walk a few hundred feet (depending on the campsite you choose) to a stream with good enough flow to easily capture water and stay dry.