Our family has been cooking turkeys outside for the last couple of decades. Here's what we do -
First off, make sure your turkey is COMPLETELY thawed. The Sunday before Thanksgiving, I make sure the body and neck cavities are completely emptied out and set the turkey in a pan in the bottom of the fridge to continue defrosting.
On Thanksgiving morning, we rub it down inside and out with olive oil and Parkay margarine, then we salt and pepper it. We do not stuff it so that it cooks really well all the way through.
We set up a 22.5" Weber grill for indirect cooking, using charcoal corals on two sides. We put a pan with a little water in between the charcoal grates with a little apple juice to catch the drippings and keep things moist.
We add a small chunk of apple or cherry wood along with a few charcoal briquettes to each side about every hour or so.
A 20 pound turkey takes about eight hours, depending on the air temperature, wind, etc.
I go by the pop-up timer for doneness. I've checked it with a temperature probe before and it was right on the money. I think that's fascinating that they can build a cheap little plastic and spring thing that is so accurate.
Hope this helps!