Logistically, you can't drive to Kodiak but you can hook up with a ferry and arrive their via the Alaska Marine Highway System. This you can do as a foot passenger or you can park your car or camper on the ferry and have it delivered with you. The other option is to purchase a ticket on one of the commercial airlines that serves Kodiak and then rent a car when you arrive. Neither is as simple as driving, but then again, the fishing is a couple of steps beyond what you'd typically expect at your average roadside fishery. If you have a penchant for sea-run Dolly Varden, Kodiak is your place.
Several streams have early season closures upstream from their bridge crossings.
This lets plenty of salmon make it through bridge bottlenecks to upstream spawning grounds. The dollies that follow them looking for salmon egg meals are, though, fair game. Salmon are, of course, usually the high priority items and Kodiak has them in abundance. Pink and chum salmon return to the streams and nearshore waters beginning in July followed by silver salmon which will steadily roll in until October. Miles of road follow the island's coastline and fishing to the salmon while they are still in saltwater is as simple as pulling the car over and wading into the surf.