We make a lot of pizza. There are very many ways to make dough, and if you do an internet search, you will find very many ways to make grill pizza, too. This is what works for me.
Preheat your grill to a real high heat. If you have a thermometer, you'll want it at about 500 degrees.
Roll out your pizza dough (any pizza dough will do, there is a recipe below) - a smaller circumference is easier to work with.
Put the crust on the grill for 60-90 seconds.
Using a metal spatula (or a pizza peel if you have one), take the dough off of the grill and put it cooked side up on a cutting board or baking sheet. I don't put any flour or olive oil on the board/baking sheet and don't have any problems. The other pizza maker in the family likes to put flour on everything and swears it helps with sticking.
Reduce the heat of the grill to medium-low - about 350 degrees. Put toppings on your cooked side of the crust while you wait for the grill to cool down a little. If you are working with a charcoal fire - I don't really know what to suggest. I've heard you can do hot side/cooler side, and you can play around with that. Really, I don't know - May the Lord Be With You.
Now, carefully shimmy the pizza, uncooked side down, onto the grill. Put the cover down and cook for 5 minutes, checking occasionally. You want to have the cheese melt and the dough cook through without burning the bottom.
Remove your pizza and enjoy.
*** If you are making several pizzas - the dough amount below will make 3 small thin-crust pizzas - you will want to do the first side of all three first before reducing the temperature of the grill. It's just a smoother process this way.
My crust "recipe" (often I just throw approximate quantities in the bowl and adjust until it has the right consistency - just a little bit sticky)
1 cup warm water
2 tsp. rapid rise yeast
1.5 cup whole wheat flour
1.5 cup white flour
1 tbs. olive oil
1.5 tsp salt
I use a dough hook and just through the ingredients into the mixer and let the hook do the work. You could also knead for about 5 minutes by hand. Allow to rise at least 1 hour. Roll out on a floured surface and make pizza!