I prefer it on a charcoal grill. You can use other grills, but I think charcoal gives you the best flavor. You can either marinate it or rub it with seasonings and olive oil to keep the meat from drying out. I would also recommend leaving it whole or only cutting it into thirds. Making the pieces thinner or smaller, by cutting it into butterfly fillets or medallions, you are making the meat too thin and it is more likely to dry out.
The trick with backstrap is cooking it over direct heat for about 2 to 3 minutes per side to trap in all the moisture. You will know when it is ready to flip when the meat does not stick to the grates. You want to get a good sear on the outside, but you do not want to overcook the meat. The meat should be medium-rare, or more on the rare side if you prefer. Once you have all the sides seared, keep it on indirect heat (this can be achieved by putting all the coals on one side of the grill when starting the cooking process), until the middle is a nice medium-rare.
You could also use a cast-iron skillet if you do not want to or cannot grill it. Cast iron is best because you are still able to get a nice sear on the outside.