Peak Season: June - September
But it's open year round
There are over a dozen campgrounds at Olympic National Park with ten of them being open all year. The sites are priced anywhere from $8 - $12.
I spent a solid week at Olympic National Park. A considerable amount fo this time was spent at the port Angeles Library and I wound up cracking one of my wheels as I raced down Hurricane Ridge (but the ride was still worth it, so much fun going down), but the park has a lot to offer. You can see everything from temperate rain forests (Quinalt and Hoh), to Glaciers, beaches, waterfalls, and hot springs.
I would have to say my favorite part of the park was the coastline. I think it's probably not as beautiful as the Oregon Coast, but it's very nice. Hurricane Ridge is pretty great and I paddled down the Hoh River which was a blast, but the Hoh Rain Forest itself was not as hot as I had anticipated. ALso if you have a teenage son or daughter you pass through Forks, WA, the home of Twilight Insanity, on your way to Hoh from the North or to Hurricane Ridge from the coast. The scene in Forks is so preposterous. Tour buses are carting giggly girls around landmarks from the movie and every business in the city is fighting for bragging rights of rights of being "your Twilight Headquarters."
One minor downfall if you plan on spending a considerable amount of time in the area is that none of the campgrounds have showers. I took one dip in a lake to clean up and I paid for a shower at the local KOA another day.