A caribou hunt is along the lines of an antelope hunt in WY. You'll either be covered up by them, or you won't see any/many. It all depends on where and when you go. Some areas just don't hold much game, others are corridors, and some areas only part of the year. Alaska is a HUGE state, with varied terrain, and many different caribou herds. The herds winter/summer range can vary greatly from year to year.
If I had to chose "when" it would be in early/mid September because they'll be bunched up by then, and the leaves will be dropping... Where, that's up to you.