Vegard, A non-resident, either out of state or from out of country can hunt in Montana on their own without a guide. The hunting on public land out here can be very good, that's where I do pretty much all my hunting. But I think for what you're wanting it might be the best route to go with a good outfitter. The logistics of packing your supplies, food etc. can be bad enough from out of state but from overseas would be tough. Plus drawing a tag in the general draw for non-resident can be tough where as there is a set aside for outfitters, they cost more but it's pretty guaranteed.
I checked on some of the outfitters that I knew, but I don't know, the Northern Yellowstone herd has dropped from aprox. 19,000 down to around 6,000 so the early back country hunt isn't what it used to be.
I'll bet Ripp has some good recommendations though and would be worth checking.

I'll also send a little more info in answer to your PM.

By the way, Hey Ripp, I think bears already opened on the 15th didn't they?