It's not a perfect metric, but 86% of people accessing publicly-financed homelessness services in 2016 reported that their last residence was located in King County.
Seattle has always had homeless people, from the time "skid row" meant the place where you skid freshly cut logs to the waterfront to be cut. Likewise, Portland, Vancouver (BC), and (to a lesser extent) Spokane have and have had very similar homeless problems of roughly the same nature. Basically, these three cities have acted as a catchment for almost anyone in BC, Washington, Idaho, Montana, Oregon, that fall through the cracks (and ya, you'll find homeless people from places like Bonners Ferry).
Basically, its the red rural areas around these cities exporting their problems to the cities that actually provide services to deal with them. Couple that with kids from these areas with little education going to Seattle to make it, and not making it.
I'd be curious to see any stats on this. How many of the current homeless were actually living in the city 10 years ago?
I'd expect that many Seattle's homeless are migrants from parts of the US with harsher winters.