My suggestion would be to first inspect the metal cap on top of the chimney and check and see if when the cap was installed, did they nail through the top of the cap. I see this a lot. The usual process is to put caulking on top of the nail heads and silicone caulk is usually what is used. Silicone is a great caulk, but not for exterior applications. The caulking will fail and you end up with leaks around the nail head.
Second, see how much over hang you have on the cap, there should be a minimum of 3" on the side. When we install caps we secure them with fasteners in the side of the cap, not the top.
It's not a difficult fix, but if you are not comfortable on the roof, your best bet would be contact a roofer or siding installer who has access to a metal break.
Oh, and welcome to House Repair Talk!