The salmon run in Alaska is a thing to behold. Unlike much of the lower 48, our rivers and streams are largely free of the developments that destroy salmon runs like dams and industrial plants. The numbers of salmon returning to Alaska streams to each year number in the hundreds of millions. Bristol Bay alone saw over 60 million sockeye return in 2018 alone.
Generally, the salmon run from May to September each year. However, it is important to remember that different species run at different times of the summer and each area and ecosystem has different timing even within the same species. Focusing solely on the more desirable species (not including Pink or Chum) the Chinook (King) Salmon generally run first, followed by Sockeye (Red), and the Coho (Silver) run last and sometimes go deep into the fall.
Also - it's possible to get a "winter catch" any time of year if you charter an ocean fishing vessel. The salmon run is only when the salmon run up the streams and rivers to spawn and die, so the rest of the time they're out swimming in the ocean and you can troll for "winter kings" in places like Homer.
If you are planning on fishing for salmon during the run in Alaska you should understand the timing is always a little unpredictable and, if you are planning on fishing a very specific area, the window can be very narrow. For example, if you want to visit Homer and dipnet across the bay in China Poot Bay, there is only a 2-3 week window, tops, when the Sockeye are really running thick there.
Even if you aren't planning to fill your freezer, it's still worth witnessing the salmon run. Streams full of fish are a sight to see and the hundreds of millions of fish are a key component to sustaining life in Alaska. They feed the people, the bears, the eagles, and a huge amount of sea life including bugs and plants!