Remove the arms, back and base from the chair. Your model may use screws or bolts. In either case, simply remove them and set them aside in groups with the piece that you remove so that you know which hardware goes with which piece.
Lay the cylinder and base on their sides. Remove the base from the pneumatic cylinder. As with the arms, base and back, there will be screws or bolts. Set this hardware aside in a group so that you know which hardware goes with which piece.
Attach the replacement cylinder to the wheel base. If new hardware came with the replacement cylinder, then use that hardware in place of the old hardware you removed. Once the base is solidly attached, attach the seat and test the cylinder to be sure that it raises and lowers normally.
Attach the back and arms onto the chair. Recycle the old cylinder and any leftover hardware.