You are likely to see moose anytime and anywhere in Alaska keep your eyes open!
We normally see bears once or twice a week on the river usually more in the fall
Caribou can normally be found roaming the peninsula in the summer they are most often seen in Kenai area near the airport or out on the flats.
Seeing Lynx is rare although there are plenty around they are very crafty at not being seen! Unless of course you catch one in the river :P
Bald eagles are found almost everywhere on the Kenai Peninsula and are seen daily on the river.