Summarize:

Use plywood, particle board, or MDF to create a base for the scratching post. Buy a piece that’s 2 ft by 4 ft by ½ in (.61 m by 1.2 m by 1.27 cm) or cut it to the correct measurements using a table saw. Exercise caution when cutting to ensure no one gets hurt. Try to pick a natural, non-treated wood. Avoid wood treated with chemicals, as these may be harmful to cats. Your carpet should be at least 3 ft (.92 m) by 5 ft (1.5 m) to ensure you have enough to wrap around the base and staple it tightly. Use an X-acto knife and a ruler to ensure you have a neat, straight line. Choose a stiff carpet, such as Berber, for a longer lasting post. Flip the carpet upside down and lay the base in the middle of the carpet.  Draw a line straight out from each side of the base to the end of the carpet so you have squares extending from the corners.  Draw a straight line ¾ in (1.9 cm) from the corner of the base that intersects the squares you made.  Cut across the straight line first, then cut along the lines you made extending from the corners. Staple the carpet to the bottom of the base on one side, putting staples in every 2 in (5 cm) using a ½ inch (1.3 cm) crown stapler. Pull the carpet tight and staple it on the bottom of the opposite side, putting staples in every 2 in (5 cm). Repeat on the remaining two sides, making sure to staple all the way to edge so your corners are flush.
Cut or buy the base wood. Cut the base carpet to size. Cut notches for the corners. Cover the base in carpet.