The only way to ensure a dry chimney breast is to make sure that rain cannot penetrate down through the stack and into the chimney breast below. Yes, it is that obvious! Bricks are pervious and will absorb water and that water must be stopped from travelling down the chimney stack into the chimney breast.
The usual way to prevent water entering the main body of the property is to install a lead tray(s) within the chimney stack. This is dressed across the stack and turned up inside each flue. This ensures that any water drenching the stack is ejected through the weep holes designed for this purpose. The problem is that older properties are often built without lead trays, though some may have a slate layer which can help. To retro fit a lead tray means that the chimney stack must be taken down to the level where a lead tray is needed and then rebuilt.
On occasions it can be that it is only the lead flashings that need attention or maybe the flaunching on the top of the chimney. These together, perhaps, with additional pointing can make a significant difference but cannot be guaranteed to fully solve the damp problem.