This is going to take either incredible luck or skill. Grizzly's are only in the Northwest region which is the final region unlocked.
Actually hitting the heart is the hard part. It is usually obscured by the bears leg and it sits at the lower half of the bears chest if its on all fours (about the same height as what you might call it's elbow).
Now depending on your position compared to the bears, you will have to angle your bullets trajectory to hit the heart while it travels through the body. If you can get directly in front of the bear and cause it to stand, you might have an easier time figuring out the location since it should be right in the middle if the chest without too much guessing. The heart is a hard target to hit since it's relatively small. Patients and persistence are key here, keep trying.
You should be using a higher powered rifle like the 30-06 or the .338 Mag to improve your penetrating power and actually be able to reach the organ.
Got this with the luckiest shot ever!
*** Spoiler - click to reveal ***I got this out of sheer luck so don't be discouraged if you are having a hard time. The first Grizzly I came across I thought this would be easy, I lined up and shot... too low. I hit the bear but didn't even hit a rib (barely grazed it), so it started to run off.
So, I decided to just try and shoot it enough time to just drop it and track it down using the blood trails.
I pointed at the running bear and fired a second shot, LUCKIEST shot in the world. As its in a full sprint away I nailed it! Rib, Right lung, heart, heart, Left lung, leg. Dropped dead right there... Can you imagine the gear credits? I didn't even know you could get two heart shots... Unfortunately I don't remember how may credits it was total but it was without the 10x multiplier which would have been even more awesome.
feel free to share your winning shot in the comments.