How do I build an "internal frame" for a reclaimed wood door?
That's a hard thing to google. I can offer 2 possible approaches:

1.) I think you could do something like these doors with battens:
 You would have to turn the battens 90 degrees from the drawing since your facing boards are horizontal. The most important part would be the vertical battens on the left and right side. This would effectively form a frame with the top and bottom boards.

2.) If you want the frame to be truly internal, you could build a mortise-and-tenoned rectangular frame, and then apply facing boards to both sides. For more info on this, I would google "frame and panel doors". You could do the simple rectangle, like cabinet doors, and leave out the panel. This frame then  could be sandwhiched between layers of facing boards. If you don't feel like doing mortise and tenons, substitute loose tenons, biscuits, dowels joints, pocket screws, etc.

Either way, screws or nails should be fine for attaching the facing boards to the frame. If you use nails, hammer them in at slightly varying angles to prevent the boards from pulling straight out.

Also, leave a little space between the facing boards. Here in the US, I would use a dime as a spacer, but you could use a couple business cards, or a small drill bit, or measure a millimeter, etc. This allows the wood to expand and contract with temperature and humidity changes.