Turn the chair on its side so you can access the bottom, where the seat attaches to the rest of the frame. Flipping the chair upside down and leaning it against the desk can make it easier to access the spot you need.
Remove the seat from the chair frame by disengaging the clip and washer that hold it in place. On some chairs you may also have to remove a screw. Set the washer, clip and screwdriver aside and carefully pull the seat away from the rest of the chair. Watch for any small parts that come out and put them with the screw, clip and washer.
Place a pipe wrench on the main shaft of the chair and tighten it down. Turn the pipe wrench to loosen the shaft and allow the hydraulic cylinder to slide out. Insert the replacement cylinder in the housing, reassemble the chair and test the performance of the height adjuster.