There are different salmon runs at different times of the year. On odd number years (like this one), we have very large runs of pink salmon that probably offer the best catching/viewing opportunities (although not the best tasting of all the salmon). That run tends to peak in mid August and occurs on many local rivers. Later in Sept/Oct. is the peak of the coho run. After that we have chum salmon in Nov/Dec. Chinook come in about 3 waves in early spring (end of April/early May), summer (around June) and fall (around Aug/Sept).
For pure viewing and rapid catching in the rivers, I'd recommend the pink salmon runs. For best eating, I'd recommend the spring Chinook in the Columbia River.