In early June the only salmon are the kings, so not likely.
The typical high points are Homer and Seward, as mentioned above.
The road system is such that there is little or nothing to be gained by using a car and hotels, and the economics favor the RV, especially in the smaller sizes. Motel/hotel availability can be an issue, maybe not too bad in early June
Joe b's suggestion of the Valdez-Whittier trip is a good one, a day to Valdez and sort of a day to Whittier, so 2 days total.
I will add going to the beach at Deep Creek or Ninilchik or Anchor River. There are beach camping areas at all. They use log skidders to launch and retrieve 30' charter boats off the beach. I am fascinated, but maybe not your thing.
Hope is interesting and a bit funky. The salmon fishing is very late June and July.
Going north from Kenai to the Captain Cook Recreation Area, there is a parking area on top of the bluff. If the tide is running it will give you an appreciation of how much water moves in the inlet. At low water there may be agates to be found.
The Kenai Visitor Center has some informative historic exhibits, and on Saturday has a craft market.
The Sea Life Center and Exit Glacier are in or near Seward, ample RV parking. There are also marine tours.
Have a great trip.