A key element to grilling a turkey is to use a drip pan. A drip pan does two things. First, it helps maintain a "cool zone" over which the turkey can sit, which helps prevents flare-ups and allows the turkey to cook without getting charred and overdone on one side. Second, it catches the delicious juices that will drip from the turkey, allowing you to make gravy despite the lack of a roasting pan.
A disposable aluminum pan works well as a drip pan, or you can cover a metal baking pan in foil, which will keep it from getting ruined by the blackening effects of the fire.
Add about a cup of liquid to the drip pan. Water is fine, although broth, some white wine, or even a bit of port will result in a very richly flavored gravy at the end. If you don't plan on making gravy, just use water, since the liquid won't help flavor the grilled turkey itself at all.