From top to bottom:
2- The liner needs to be in good shape. Check the part that is visible from above, and occasionally have a chimney sweep check it all the way down. Some sweeps have special video cameras that they can lower down and you get a video recording of every inch of the inside of your chimney where you can see cracks, missing mortar, and creosote build-up.
3- The chimney cap is the concrete "hat" that joins the liner to the brick. It must not be cracked or it will let in damaging water. And it should overhang the brick and have a little groove all around the underside. This groove ensures that rain drips straight down from the underside of the cap rather than flowing back along the bottom surface of the cap to the brick.
4- Check the condition of the brick and the mortar. White staining is salt coming out of the mortar, indicating a lot of moisture movement. That means either a leak from the top, or a lot of humidity from the house coming up the chimney, through the liner, and into the brick. In either case, although the staining is not itself a serious problem, it?s a sign that you need a moisture control plan. If the mortar is breaking up, you will need to chisel out a little mortar and replacing it with fresh mortar, called ?repointing?.
Look up "Chimney" "Flashings" in this database for more details about how this should be arranged. It is all too often done wrong. A single piece of metal sealed to both the roof and the chimney will not last.
7- Study the effect that exhaust fans can have on the draw in your chimney. For more information on this check out "Spillage Susceptible"