The Olympics are very good for backpacking. I'm not much of an expert on the Olympics, so I would suggest you get the following book: http://tinyurl.com/43aa73
and/or, visit www.nwhikers.net. NW Hikers.net is a great place to ask questions like this.
The classic rain forest hike is up the Hoh. You can go from the end of the road until you get to glaciers underneath Mount Olympus. You will probably see people around the parking area, but once you get in a few miles, I doubt you will see too many (especially this time of year). If you want more solitude, there are similar river drainages that parallel the Hoh (Queets, I believe).
For beach hikes, I like the Shi Shi area. There are a bunch of other hikes, though, that allow you to get way in there (the Olympics are a land of really long valleys, with far fewer lakes than the Cascades).
When you post the question on NW Hikers.net, I would post it as you did here. There may be Cascade hikes that incorporate the rain forest along with the mountains. The West side of the Cascades is plenty wet (although not quite as much of a rain forest as the Olympics). You may experience that wetness first hand. Today's forecast: http://www.atmos.washington.edu/data/zone_report.KSEW.html
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I don't think there are any hikes that take you from the beach up the rain forest to the mountains, without crossing a road (or even walking the road a bit). It may be, though, that it would only require a bit of road walking. Such a trip would make for a very good story (beach to rain forest to mountains). Also, there is a trail from the Hoh valley trail which allows you to go up and get on High Divide (which can be spectacular) so you have more options that way as well.