Seattle is a young city with a long memory. The Duwamish people lived around Elliott Bay for thousands of years before the Denny Party landed at Alki Point in November 1851, and the city took its name from Chief Seattle, a leader of the Duwamish and Suquamish. Then came timber, a fire that burned the whole business district in a few hours in 1889, a Klondike gold rush that made the merchants rich in 1897, a world's fair in 1962 that left behind the Space Needle, and the companies that changed how everyone else works: Boeing, Microsoft, Amazon, and a coffee shop that opened in Pike Place Market in 1971.
It's a city of water and hills, with a ferry terminal, two lakes, a canal and a snow-covered volcano that shows up on clear days and disappears for weeks when it doesn't. Honestly, the rain is part of the deal: the airport records measurable rain on roughly 156 days a year, mostly between autumn and spring, and July and August are the dry months. Five days is the right length. It gives you the famous things without rushing, a ferry ride, a day on the mountain, and a couple of afternoons to sit with a coffee and watch the weather change.
This plan keeps each day to one part of town. Day one is Pike Place Market, the new waterfront park and a ferry to Bainbridge Island, day two is Pioneer Square, the International District and Capitol Hill, day three is Seattle Center, day four is Mount Rainier, and day five goes north to Fremont, the Ballard Locks and Discovery Park. We recommend starting on a Wednesday. That puts the mountain on a Saturday, when the visitor center at Paradise is open in winter (it opens only on weekends and holidays), and the locks on a Sunday. If the Saturday forecast is poor, swap days four and five. Seattle is also one of the best bases for the United States northwest corner, so the More Time section covers the islands, the peninsula and Vancouver.