Gutted and gilled with the head off, of course. My grill is a standard round Weber. I put the coals on either side and make sure their really hot (use compressed air on the coals before the fish arrives.) The grease runs off into a pan of beer. The beer provides a little replacement moisture.
The Michigan state record lake trout is 61 pounds. I've seen (but have never taken) a few forty odd pounders. My grill is pretty good sized (20x24), but even without the head and tail I'm pretty sure I couldn't fit one of those monsters from corner to corner.
Cooking lake trout taken from Lake Superior, IMHO, is strictly optional. Cure filets for a couple of days in the fridge with some salt, brown sugar and spices, slice thin and serve on rye bread. Yum!