A chimney chalking may expose a leak. Chalk near the chimney could be moisture entering the masonry above the chalking area and pulling lime or ‘salts’ out of the mortar leaving a white residue, like salt crystals, of caulking on the roof. There are several common sources of water getting into a chimney.
It could be from the masonry cap formed at the top of the chimney that tapers from the edge to the flue to drain water off the chimney. It is not uncommon to see a relatively new cap with cracks that allow rain into the core of a chimney.
Sometimes soft mortar and/or hairline cracks will weep water into the chimney.
Metal flashing where the roof touches the chimney could have been done poorly.
The moisture on the fascia is trouble. It will cause rot sooner or later. You must find the source of water and if it is in the flashing, or lack of, do not tar it. Flashing is designed to breath. If installed properly, water that gets under one area of flashing will be directed out by the next layer below. Tar will only trap water and cause leaks.
The chalking should come off with a 1/4 cup TSP per gallon in a warm water solution. Apply the mixture twice, about 5-minutes apart, and see if the chalking softens. If not, apply a 3rd application. Don’t rinse between these applications. Or you could try cleaning the chalking with oxygen bleach. It is just a different chemical and milder.
Some chalking from paint requires a solution of muriatic acid but I would save that for a following weekend.