Disassembling and cleaning wood stove flues and chimneys is dirty work. Make sure you cover nearby areas with old blankets and thick plastic sheets taped in place before starting. You need to ensure you have access to the stove doorway and/or the flue pipe -- but try to make as little mess as possible.
Stove pipes with an access door
Open the stove pipe access door. These are sometimes referred to as soot doors. Shine the torch up the pipe. Ascertain there are no blockages, challenges to structural integrity or other concerns, such as rain penetration. Deal with any issues first before starting to clean.
Insert the head of a chimney brush with a diameter slightly larger than the diameter of the stove pipe. This will ensure the brush engages firmly with the inner surface of the pipe and removes plenty of soot. Brushes made of wire, polypropylene and polymer are suitable.
Push the brush rod upwards into the pipe, leaving enough rod dangling to allow you to connect the next rod. Screw the next rod in place. Push the rod upwards as before.
Continue connecting rods and pushing them up the stove pipe until you feel the brush has popped out of the top of the pipe and entered the chimney cap, if present. Take a firm grip of the bottom rod and pull it so the brush re-enters the top of the pipe.
Pull the rod all the way down the stove pipe. A huge amount of soot will probably be dislodged. You may have to do the process more than once if the stove pipe has not been cleaned for over a year.
Stove pipes without an access door
Photograph the stove and flue from several angles to create a record to help in later reconstruction. Remove hinged stove doors to gain better access. Also, take away any detachable adaptors, twist-lock anchor plates and tee plugs in the system if necessary.
Pry open the pipe lock tabs, where present, with a screwdriver. Untwist and remove the pipe components, if they are of the Selkirk flue type. Fixed pipes and chimneys cemented in place with fire cement or other jointing compound may have to be broken apart to clean fully, depending on access issues. Tap the cement lightly with a pin hammer to crack it. Brush away the pieces and dust.
Remove the components and take them outside to avoid making a mess inside. Clean them individually with a chimney sweep brush, by pulling the brush backwards and forwards within them. When they are de-sooted, return and re-cement them back in place.
Use an industrial vacuum cleaner rather than an ordinary domestic type to clean up.
To avoid all the trouble and mess associated with doing the work yourself, employ a professional chimney sweep. Prices start at around £40.
Wear a soot mask and goggles to protect airways and eyes.