How to build a brick oven bbq pit was actually answered in one of those questiona dn answer places as a question about how many bricks would be needed to build. Although it is true there are many people who build brick ovens with actual bricks this is usually very ineffective and much less efficient than the cost and labor might imply. Look at this page for a much more reliable alternative.
We built this pizza oven but this is what some people may call cheating because we bought the pizza oven and then built the brick structure to hold the pizza oven and to make the proper architecture more reliable because the appliance has been tested and certified as a stand alone appliance.
To build one of these your self you would basically need to re-create the structure without the benefit of experience or inspections to guide your effectiveness and safety. There is a lower box for the wood burning fire and an upper box that is where the food or pizza will be inserted.
If this is to be a barbecue grill we only need a grate inbetween the 2 boxes so there is a stone lined lower fire box with a grate above it and then a upper box for the food to sit on the heat radiating from the flames.
For a pizza oven we need a stone liner in the firebox so the heat does not break the outer brick structure with vent holes along the stone divider above the flames. This stone divider is the floor of the food-box and the food box stone should line the entire area because a pizza oven is designed to heat the stone so the stone will radiate heat for several days. A pizza oven uses heat much more effectively than any barbecue grill because a small wood or charcoal fire is enough to heat up the fire brick liner and the fire brick will radiate heat to the food for days.
Most masons who custom build pizza ovens use an insert that is made of high heat ceramic or carbed stone like the one in this image. This is an outdoor kitchen we built. The home owner purchased a pizza oven insert but placed in on the stand instead of building brick masonry around the dome. Usually the only part of this insert that is visible is the opening where the pizza is inserted with a giant spatula.
This kind of domed pizza oven holds the food and the fire under the dome and the dome gets very hot and holds the heat for days.
With a brick barbecue grill we would build more like the structure above that has a separate box for the fire and a box for the food. The only thing not obvious is making sure the fire box has adequate ventilation so the fire has access to fresh air. Wood and charcoal need vents to control the heat and to continue burning.