Wow, great question! It's not all that bad though, and being summer (YEAH!) now is the right time to be looking- as everyone is euphoric. So here's my advise:
1. It's numbers game, so here's what you do: sign up for a kick-ball league, or if you love sailing go down to the CYC (Corinthian Yacht Club) and tell them you want to do committee boat, or volunteer at something you love doing, hell join a flash mob- those guys practice a lot so there's lots of opportunities to meet folks.
2. The other key to making friend's in Seattle is coming into contact with people that YOU are really excited to hang out with.
Not a difficult formula, but it takes some initiative and a bit of forethought.
One last thing- I can tell you that you probably won't have much luck going out dancing, or hitting bars in Seattle. For some reason those venues seem to be almost exclusively for hooking up or hanging out with friends you already have, more so than anywhere else I've been.
Cheers
Good luck!
Sean