Depending on the age of the deer the backstrap may be tender enough to grill as steak. My favorite way to eat deer meat other than sausage or as hamberger meat is in a stew with onions and peppers served over rice. Depending on what you like it could have anything from carrots to tomatoes in it.
Sorry I dont follow recipes too well so I normally just wing it when I'm cooking.
Not a crock pot for back strap! That's a waste unless you have plenty. If you want something along those lines, strip the strap into small pieces and make a stroganoff type dish.
Other than what you already mentioned, I think grilling them over a pit is an excellent choice also. Medium rare! If you want to jazz it up a bit, make a nice red wine sauce from a demi-glace.
If, as WB said, it is from a large or older deer, marinating it to tenderize it may be in order. However, marinating usually masks the flavor of the venison. If you can just get a kind of meat tenderizer that has the long needles, that is the way to go. After all, why eat venison if you don't want to taste it! Go buy a cubed steak instead, same ****!
I always eat at least some of it as soon as I dress out the deer as kind of a celebration. Usually at camp so good to keep it simple with few ingredients. Marinate in orange juice and garlic powder. Poke small slits with a filet knife and stuff each with a sliver of garlic. Salt and pepper generously. Put on a HOT grill and just sear on all sides. If you let it overcook for even a second it will turn too tough to eat. Let rest and slice into medallions. Throw some new potatos, onions, etc. on the grill and you got a slap yo mama meal in the woods.
Other than what has already been mentioned we recently tried cutting it into 4" strips about 3/8" wide. Marinatted in allegro and cooked down in a skillet with taco seasoning. Makes awesome tacos and burrito's ! You can also use the tenderlion from inside the back of the deer also. The key is do not over cook it as deer meat is very lean. Also be sure to clean the talo and any membrane from the meat.