Though I spend a lot more time deer hunting, caribou are my favorite game. I live in the part of Alaska that is thin on caribou and rich in blacktail deer...
Caribou are hunted in late summer/early fall when the tundra is turning bright orange and red. Depending on how far north you go, you still have 18 to 20 hours of daylight and the time of year is (generally) dry and pleasant with temps in the 60's - it's beautiful.
You might spend days glassing the rolling country without seeing a Boo, then suddenly the entire horizon will start to move and you may see thousands trotting past within a few hours. They are not a particularly smart animal but it still takes a lot of work and skill to recognize a big bull and get in for a shot in that open country - sort of like pronghorn I suppose, but time is limited because these animals are moving and once past you can forget catching up with them. You have to watch the wind and use every bit of cover to slide and slither your way into a position that your target will pass.
I think caribou hunting appeals to me as much for the country you are in than for the actual hunt itself. Where you find caribou, you will also hear and see wolves, tundra grizzlies, moose and an amazing variety of smaller critters and furbearers - and ptarmigan sometimes so numerous you can fill a bag in an hour.
It's a great hunt that must resemble the experiences of the early plainsman or even African hunts back when that country was more open and uninhabited.