Kings are usually pretty tough to get to take a fly. Both the Green, White Rivers get runs of them, but I'm not sure about the Carbon. Silvers and Chum salmon are much more likely to take a fly. Chartuse and Yellow streamers seem to work the best.
The Green is closed for Chinook, at least the last time I checked. The Carbon is combat fishing, but has kings nonetheless. You could also try the Nisqually, or better yet, the Olympic Peninsula rivers. Be sure and check regs. I only rarely fish for salmon, and even rarer do I catch a salmon. But if you really want to, try big flashy streamers and wets. But, as Steve said, Chums and silvers are a lot better fly fare.