Check your local running stores. The app Meetup can be a good place to start, too.

Check out Basecamp Cafe / Gearhouse. I don't live in Seattle anymore, but found some trail running peeps through there back in the day.

Good place for other outdoor stuff too, like backpacking and climbing.

I'm not from Seattle, I'm from Australia. But look up Parkrun.

I know there are some in America. They're everywhere in Australia, you turn up run and they track your time every Saturday morning.

They're great for meeting people

When I lived in Seattle I met a few good running friends by joining a Green Lake running group. I lived in the Greenwood/Fremont area and found a group through the Super Jock and Jill or Fleet Feet. I would go to the running gear stores and ask about a weekly running group.