Firstly you'll want to save up to buy a high powered scope as well as a high caliber rifle as soon as possible. The .7mm Mag or higher. These make hitting vital organs much easier because of their penetrating power. I found the smaller caliber weapons would bounce off rips and other bones too often.
Secondly, you'll want to get the pro hunts done ASAP as well. As once you get them done they give you a 10x bonus for your clean kills. Double lung shots give the most money at 5200 and are actually the easiest to get.
Thirdly, beat all other missions (assuming you're a completionist, if not skip this step.) Then buy all weapons and ammo types.
Fourthly, at this point you'll want to equip the .388 Mag Bolt Action with .338 Mag Polymer Tip ammo, Scent Mask, Moose Call, and the highest powered scope you have.
Lastly, you'll need to grind out some kills in order to purchase the rest of the items you'll be missing. Go to the Northwest Region and load up Open Hunt on Paddlers Bog. When you start there will be a Moose, possibly female but it doesn't matter, to your left (SW). Crouch and use the Moose call immediately. This will show you exactly where the moose are as well as get the male to walk towards you (again it doesn't matter if it's male or female, but I think the males are the only ones that respond to the call).
Then once the moose has stopped moving it will turn sideways give you a really nice shot lined up for a double lung shot. Take the shot. Should be double lung every time. In fact, if you're good enough you can actually shoot the moose 4 times in the lungs with two shots. Almost every time you should the moose it will run left before running right. So when it takes off left, just wait for it to loop back around. You don't have to lead it or anything, just make sure the red circle is lined up with the lungs. It's worth 8800 (9600 if you can get the stationary target bonus, but I'm not sure what that is exactly) rather than 5200. However, should you note that if you miss a vital organ or hit the stomach on your second shot, the payout will be less than 5200. So it's your call.