Any type of brick will suffice for the structure of a grill but firebrick is the most desirable to line the inside. Firebrick will withstand very high temperatures and is denser than most bricks allowing it to hold the heat. It will also even out the temperature if the heat is changeable. A leave-in thermometer will be a great choice to help you keep track of your food temp.
You can also use firebricks to use as a baking surface to bake flatbread, or even to make a smaller oven.