I've actually had really good results for grilled pizza, without a lot of extra equipment. I used a Weber silver genesis (the old one with grill tubes running the long way) for all my experiments. I think the grill is a great standin for a pizza oven with an intense heat source on the bottom of the oven. The heat from the bottom really blasts the crust and makes it crunchy and crispy.
method 1: thin crust pizza, lightly dressed
Put entire pre-dressed pizza directly on a pre-heated grill. Cover and blast at full heat until the crust is set and starting to show nice grill marks. Lower heat until toppings are heated and melted.
method 2: normal and thick crust pizza. Heavier toppings, pre-cooked.
Preheat grill at high, turn heat to medium. Oil pizza round for extra browning and flavor. Place undressed pizza round on grill at medium heat. The high heat start will put some nice marks on the dough, and give the dough a hit of oven spring. Grill until half cooked and grill marks are apparent. Flip and dress pizza, or hold dough until needed. You can dress the pizza on the grill, or take it off and allow your guests to have a pizza dressing experience. Works great for the kids! Grill covered until toppings melted. You can get great char and crust on the bottom of the pizza by turning the heat back up to high for a bit.This requires some pretty good timing, since it's easy to overcook and burn the crust at this point. I think it's worth it to get the browning on the bottom as well as the top of the dough, but you're going to play with it and overcook some pizzas along the way.
Also, I think stones or steels work great. I used to use a cast iron Lodge grill heated for 15 minutes at max grill temperature to cook the dough round. I stopped using it because it was just so unwieldy and heavy to move and clean.
If you want top browning on the toppings, grab the blowtorch!
