I don't think it will help much to add doors.

To avoid smoke coming into the room, you need to make sure the flue is preheated and the fireplace has a good draw before the logs burn. Usually the easiest method is to crumple up a long section of newspaper first, light the end, and hold it so it burns up very close to the damper. It makes a very hot (but short-lived) flame that goes right up and heats the chimney flue. Once you get the flue heated, then start the wood fire below, and the smoke should follow the heated air up the chimney.

Bruce

You can buy a used wood fireplace insert ( a metal box with glass doors). I use fire starter for fireplace and campfire.