Prepare a gas or charcoal grill for indirect heat. If using a gas grill, turn all burners to high and close lid. When the temperature inside the grill reaches 400°, lift lid and turn off one of the burners, creating the indirect-heat area. If using a charcoal grill, light 50 to 60 briquets and let burn until just covered with ash, 20 to 30 minutes. Mound them to one side, leaving a cleared area for indirect cooking.
Fold two 12- by 18-in. pieces of heavy-duty aluminum foil in half widthwise to form rectangles. Using the tip of a small knife, make holes in rectangles about 2 in. apart and widen each hole to the size of a dime. Grease foil with olive oil; set over direct heat for 2 minutes.
Meanwhile, rinse and dry salmon fillets. Brush skins with olive oil and set 2 fillets, skin side down, on each foil rectangle. Cover grill (if using charcoal, open vents on lid) and cook fish until skin is light brown and really sizzling, 5 to 6 minutes. Slide foil to indirect heat, cover, and cook until all but top 1/4 in. is cooked, 3 to 8 minutes. Slide fish back over direct heat, cover, and cook until fish is cooked through (cut to test) and skin is browned and crisp, about 3 minutes.
Transfer foil with salmon to a baking sheet and, sliding an offset cake spatula or other thin spatula between salmon skin and foil, very gently free fish from foil. Season with coarse kosher salt and a few drops of fresh lemon juice.