tireetim, good evening.
Several potential things?
Is the Flue Vented at room level and at the chimney pot? [but the pot vent needs to be rain protected]
You could simply introduce a barrier between the "damp spot" plasterboard and anything else you care to retro fit? BUT the dampness will be on-going under / behind the newly applied [whatever]
If the area of chimney breast is not too large, my preferred option would be to strip back the area on the chimney breast and using treated timber then Insulated Plasterboard to effect a repair.
Lets not forget, flues contain some nasty contaminants, and the big one is Sulphate, you may find that the dark marks are indeed the effects of Sulphate attack, what happens is that any fossil fuel Coal, Wood, Peat Etc. produce Sulphate, this chemical adheres to the inner surfaces of the Flue. when moisture comes into contact with this chemical, a very mild form of Sulphuric Acid is formed, this acid then attacks especially Lime Mortar and dark stains form.
You have probably seen on the Gables of older properties the dark stain mark from the Fireplace to the chimney head, a direct result of Sulphate attack.
Ken