Go to Alaska. It's cheaper and there are less hassles with guns, licenses etc..
Outfitters will run 3500-6000 + airfare and licenses.
If you're looking to spend 4k look at Adak Island. Lots of caribou, but airfare is the killer.
You could do a decent DIY hunt somewheres north of Glenallen (Just one area that comes to mind) for around $2500 bucks.
Tags will run $140 for a seven day hunting license and $325 for the caribou tag. I'd suggest throwing in a wolf and black bear tag as well.
Get your airfare early and it's fairly cheap. Under $500.
Hotel or B and B (or a forest service cabin if you get it reserved early enough) for 5 days- 250-500 bucks depending on your desired comfort.
Truck rental- 80-100 bucks a day- $500
$700-$800 for taxidermy and shipping.
Tags- $465
Car rental- $400-500
Lodging- $250-$500
Airfare- $400-$500 (If you get it early)
Taxidermy (shoulder mount)- $700-$800
Check the meat as luggage.
$2200-$2765 For a five day hunt.