For starters, what do you mean by BBQ .... a permanent brick and mortar affair, a built-in prefab ... a portable on wheels? Each of these might require a different kind of "base."

Is this a good location? Usually, one would want to locate the grill away from the house as there's a lot of particulate matter, mostly grease, that rides up in smoke that would get all over the wall and eaves. There's also the fire hazard component. We can't see enough in the picture to know if it doesn't matter or you're in for some grief down the road. Also, it could not be located near any plastic trim parts. Check to see if local code allows a semi-permanent grill installation immediately against a house or other building. Even if it does, I'd be looking for an improved location.

Can you reveal the inner construction method of the surround? The one in the picture, since it is sitting on a deck, looks like it would be light weight -- some kind of a metal framework clad in stone veneer. If it was entirely a masonry structure, it could not be sitting on a deck. If it is a manufactured framework it would be best to follow the manufacturer's instructions. If it ends up sitting on the patio floor, the cross-slope (in the direction of house wall) would be level, and the slope perpendicular to the house would be pitched SLIGHTLY to drain (1/8" per foot.)

If you're choosing the grill location by the house an account of trying to obtain some benefit of partial shelter, it would be better to move the grill somewhere away and create shelter, even if temporary (while in use only) for it.

That looks better.

If you're doing a lightweight veneer on a frame (which is what it looks like) you need to properly extend (tamped base+sand+pavers & all compacted) your paver patio to the wall and it will support the surround and the grill. If you're building a masonry surround, you'll need a concrete footing.