Regional discussion and conditions reports for Washington and Oregon. Please post partners requests and trip plans in the Pacific Northwest Climbing Partners section.
I'd go August/September. It can still be rainy in July and buggy. By August the bugs are dead and the weather is more stable. The rains don't generally come until October/November so September is usually great.
While Sept. can be spectacular,(if you can come then), I'd go with mid-July into Aug. for the trip you've suggested. Also spectacular, especially as the snows melt up high.
From the middle of July to the middle of September the weather in Portland and Seattle tends to be fair--morning clouds in Seattle. On the Oregon coast summer weather tends to be cool unless there is an east wind blowing--often in September. At Mt Rainier most years the snow level would be ok for hiking around the end of July--give or take depending on the winter. The problem with September is that the wildflowers are likely to be done for.