I just finished eating a piece of backstrap that was some of the best I've had, and I've been eating them for 30 years.
Try this:
Something I do first: Take a section of strap about 5-6 inches long and thaw the meat slowly in the fridge. After thawing, I let it stay in a sealed container in the fridge for about 5 or 6 days before seasoning. This allows it to age a little and makes it more tender. You don't have to do this, but it makes for a better steak.
After setting in the fridge, sprinkle a little Grill Mates mesquite marinade on it. It's a dry seasoning and if you read the package it says to mix with oil and water. I don't do that. I just sprinkle a little seasoning on one side, then drizzle a little olive oil on the strap and mix together to form a paste. Coat it well.
Flip the meat over and do the same to the other side. Let it sit in the fridge over night.
When you are ready to cook, get your grill as hot as you can. I use a product called Grillgrates to assist in searing the meat. I have found chicken, venison and vegetables all come out much better with these grates than when I cooked them on the regular grill.
When the grill is HOT, place the strap on it for about one minute, lid down, then flip for another minute to sear both sides. After that, cook to the desired doneness, but anything over meduim really ruins a good cut of meat, and backstrap is no exception. I like them medium rare. The size of the steak will determine how long it takes to get there.
After taking it off the grill, let it rest for at least 5 minutes...do not cut it to see if it's done or stick a fork in it, remove it with tongs or a spatula. While it's resting, I grill a few skewers of peppers and mushrooms to go with it.
Whatever is left over is super tender and tasty if heated up in the oven the next day, just make sure you only heat it, and don't cook it any longer.
Best venison steaks I've ever had were done like this, as good as any filet mignon from the finer restuarants around here.