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If you have already gone to the Court's website that will give you the general overview.
To physically file the document, you need to go to the courthouse in the County where the action can take place. (This is based on "personal jurisdiction" over the defendant). This can be established as follows:
Take at least 3 copies of your summons and complaint, and the filing fee.
Once you get your complaint filed hire a process server to serve the summons and complaint on the defendant. (Any third party can do this, but hire a process server, they don't charge very much and they will get it done properly the first time, plus, if you are lucky enough to get a "default judgment" against the other person (they simply don't show up), you will want an ironclad proof of service). Your local bar association can give you referrals to process servers.
For additional pointers and tips, I do recommend visiting your local law library. (I don't know which county you are in, but some of the larger counties have excellent libraries, but even the smaller counties will all have sufficient resources for a small claims case (and the school law libraries at Gonzaga UW and Seattle are sometimes open to the public).
A process server will only deliver the paperwork - they will not prepare it for you (they usually will prepare the "proof of service").
Washington does permit service of small claims suits by Certified Mail, but I don't recommend using this method because if the defendant fails to show up and you get a "default judgment" (meaning the court grants you a judgment for the full amount you are asking for without having to go to trial) you run a risk of having your successful "default judgment" attacked by the defendant later. Having a proof of service from a process server reduces (not eliminates) this risk.
Here is another helpful publication that may assist you with your matter: http://www.washingtonlawhelp.org/files/C9D2EA3F-0350-D9AF-ACAE-BF37E9BC9FFA/attachments/39285D5E-B302-1D9D-41E3-E3BF638D39B7/9940en.pdf