Decide on which version of the captain's chair to build. The show started with one model of the chair, but creators developed a new chair following the ship's refit after fighting the Xindi. There is more information available about the first chair, because it was used for the first three seasons of the show.
Watch episodes of the show. Keep a close eye on the captain's chair and take notes on every aspect from color to accessories. Different episodes featured new aspects of the chair, such as a retractable panel on both the left and right arms.
Print out pictures of the chair at various angles. Take the pictures to a furniture store and find the model of office or captain's chair that looks most similar to Archer's chair. You won't find anything exact, but you can find a close approximation and make adjustments yourself.
Order the chair in the appropriate color, or remove the fabric from the chair. Visit a fabric shop to find an exact color match and reupholster the chair using the new fabric. Staple the new fabric to the chair, adding padding to mimic the shape of Archer's chair.
Measure the dimension of the two arms. Cut plywood pieces to fit around the arms and glue them together. Drill small holes for LED wiring and a plug for the LCD screen. Measure the LCD screen and cut a hole in one of the arms to fit it. Paint the plywood to match the pictures of the chair. Install the lights and LCD monitor in the arms.