Chimney may be leaking - what should we do?

[Virginia Jones]

We recently finished renovating a rental property and after a heavy rain it looks like the chimney may be leaking. It is not an active chimney. There is actually not a fireplace but at the base of the chimney in the basement is a little door with ashes in it. There are black streaks running down from the door and after a heavy rain, a little pool of water on the basement floor. 

Should we have someone come out and inspect the chimney or is there an easy fix we could handle ourselves? The chimney does have a chimney cap on it.