Grilling on an iron fire pit
This can easily work as a grill.  A cover is not necessary. You simply need a grate that fits over the surface. I would go for a grate that fits over the rim, you don't want to worry about shifting and balance while cooking.  Holes in the bottom for air flow would be nice, but again, not necessary.
Start a fire, burn down to coals.  You can use the &quot;hand test&quot; to check the temperature.  Grill your product.
In a grilling situation, the flavor is initiated from drippings hitting the coals, it is not necessarily a smoking situation, though you will pick up some smoke depending on your fuel.