The season starts in early April with king (chinook) salmon, which runs until late July. Sockeye comes into season next starting in early June and only running until late July.
Pink salmon's season is from late July to early September, coho salmon's season is from late July to mid-October, and Alaskan chum salmon is from late July to early September.
The exception to this rule is a small number of salmon fisheries that operate year-round. It's hard to know if these fish are fresh or not, so it's best to opt for fresh-caught.