Im a Mason contractor in Virginia Beach, Va. We deal with alot of leaking chimney's. It may be that you need to completely remove the shingles back from the chimney and apply a sheet membrane around the chimney install the counter flashing and make sure the step flashing is turned into the motar joints and not simply pasted to the chimney and caulked. using a multi-flu chimney cap will help but make sure to top cap is extended approx to the outside edges of the chimney chase walls. these can be special ordered if needed by a chimney sweep company. Also make sure you seal the chimeny with an industurial grade sealer. I dont recommend thompsons. Their is a product call Chim-saver that has a 10 year guarantee. This is supplied buy copperfield chimney supply. you should be able to find them on the web. Make sure their is no cracks on top of the crown. and that you seal the top of the crown as well. Do the most obvious thing first the move forward. you can be sure the problem is not related to chimney from the roofline up if you wrap the chimney from the roofline up with plastic and wait for the next hard rain. if it does not leak you know it is related to the chimney somewhere from the step flashing down. if it does not leak the leak is somewhere from the roofline upward.