What does water leak through my chimney and wood stove pipe?
There should not be anything dripping from your liner's insulation.

Your local chimney sweep should know how to inspect your liner and its top termination to determine if it was properly installed to conform to local code, including the installation instructions for that liner product. The space surrounding the liner should be covered (metal or masonry) to prevent entry of rain, snow, birds or other small animals, whether or not the liner itself has a rain cap or spark arrester. If your liner was required to be insulated (e.g., for purpose of helping induce a draft more quickly), it won't work very well if the insulation layer is soaking wet and then freezes.

You might browse through NFPA 211, a national standard for "Chimneys, Fireplaces, Vents and Solid-fuel Burning Appliances", and the installation manual for the stove and for the chimney (if it's a factory-built liner).

When we recently had a liner installed for a gas-fired log (not the same, I know, but offered as illustration), the chimney kit included a cap device to support the new pipes (exhaust and combustion air) and to block off the top of the old flue. It was simply fastened to the top of the old chimney and the rain cap added to the new pipes.