Chimneys are actually quite easy to build, with all parts available at your local hardware shop. It's more like building a Lego block kit than anything else, but it still needs to be done right for your safety and protection. Always contact your local fire brigade to make sure that your plans meet all zoning and building regulations before you begin your project.
Estimate the direction you'll need to travel from your stove to the exhaust hole. You will have to decide this first before you even buy any stove pipe so that you know what type of pipe you will need.
Start laying your stovepipe. Begin at the building exhaust hole and work your way down. Lay down a bead of mortar caulk inside the hole, then take the first pipe and push it into the exhaust. Then, using flex pipes and straight pipes, make your way down to the stove.
Create a seal with the clean-out pipe. Where the stove exhaust hole is, lay down a bead of mortar caulk inside. Then, push the clean-out pipe into the exhaust hole. The mortar will make a positive and smoke-proof seal.
Connect the clean-out pipe to the rest of the chimney. Your clean-out pipe will have a removable bottom that sits below the flow of smoke. This is a very essential piece on your chimney, as bits and pieces of creosote or unburned wood will collect in here and not clog the pipe.
Secure the pipe connections. Wherever there is a connection from pipe to pipe, drill in three sheet metal screws to secure them together. Make sure they are equidistant and tight.
Call your local fire brigade and ask for an inspection. This is important because building your own chimney, no matter how you do it, must conform to local fire code ordinances. Even though you checked with the fire brigade before you built the chimney, you must also have the fire brigade approve it afterward.
This is a fairly simply project, but it does involve working "through" a wall. Therefore, you should do this while you are installing your wood-burning stove -- not after you have completely installed it. When buying the stovepipe for your project, get a combination of stovepipe that fits the exhaust on your wood-burning stove. Most stovepipe is either 15 or 20 cm (6 or 8 inches), so you'll need to measure the size of your outlet first before you can proceed. You'll need straight pipe, some flex corners and a clean out.
Always have your chimney checked each season before you begin using your wood-burning stove, and never leave fires unattended.