First, welcome to Oregon!
Second:
www.co.multnomah.or.us/sheriff/chlunit.htmThat's, of course, if you are actually IN Multnomah county.
Concealed carry permits are a "will issue" in all of Oregon and are very typically run by the county sheriff's departments.
You can either show proof of prior CHL ownership (i.e., your old CHL card) or you'll need to take a CHL class from an NRA certified instructor for ~$60-65. Fork over about $60, a set of fingerprints and picture (taken at sheriff's office), and in about 3 weeks you'll have your permit in the mail.
Now, a few "nuances". Most states do not "reciprocate" Oregon CHLs, namely Washington and Idaho. Thus, your CHL is not valid there. So, what most people have done is grab a non-resident Utah license in addition to the Oregon. Utah's are valid for I think 23 states total, and nearly every close Western state except for California, Nevada, and Oregon.
Some of us attended a class for the Utah permit (and also provides for the Oregon one as well so only one class was needed). Guy by the name of Roger Moore at 503-668-0417.
Of course,
www.packing.org is also a good resource.