One way is to head north and pick up the Northern Tier to Whitefish, MT. Then due south thru Seeley Lake. Not the most direct, but very scenic.
+1. You can take a ferry out of Seattle to pick up the Pacific Coat route. That takes you to just east of Anacortes, where you can pick up the Northern Tier route. Great Parks will take you south from Whitefish/Columbia Falls to Missoula. As noted, it's not the most direct, but it's very scenic. Going this way also gives you few days of warm up before you cross the Cascade Mountains. Both times I rode Seattle to Whitefish at a liesurely pace is took about 17 days including rest days in Withrop, after the Cascades, and Sandpoint.
If time is not a factor, you could also keep on the NT to Glacier National Park, pitch camp for a few days, ride up and back down the west slope of Going to the Sun, then back track to Whitefish/Columbia Falls. That would add two days.
From Whitefish to Missoula is an easy three day ride. Depending on how much mileage you are willing to do, you might be able to pull it off in two days. I left Glacier, rode back to Whitefish then to Big Fork. The next day I camped at Lake Alva. Day three I was in Missoula. Don't recall any of those days being difficult.