I've been looking around the internet and even the website to my local city hall. I haven't been able to find out how to run for a city council position. Can anybody help me out.

Contact the local city clerk's office.

DG's advice will get you filed as a candidate in most locales, in exchange for a (usually) small fee.

Have you called the local party offices to express interest? In Seattle, they are:

Republican Party
(206) 467-1996
1305 Republican St
Seattle, WA 98109

Democratic Party
(206) 622-9157
157 Yesler Way Ste 510
Seattle, WA 98104

Libertarian Party
(206) 329-5669
400 NE 45TH St Ste 1776
Seattle, WA 98105

Links to some others are found here.

I gather you may have already visited the King County Elections Site.

If you are not in Seattle then try the party offices where you live. They should be in the phone book.

Get used to using the telephone if you intend to run a political campaign!