This is my favorite Model 27 shipped in March 1960. It cleans up real nice.
I'd start by removing the grips and the cylinder.
I like Hoppe's 9 and their BoreSnake to clean the barrel and cylinder.
Always clean the barrel the direction the bullet travels. That's why I prefer a BoreSnake to cleaning rods.
After cleaning the gun with Hoppe's 9 use brake clean to rinse it off and a can of compressed air to help evaporate the brake clean.
Lightly apply Flitz with the end of your finger and lightly rub the polish to remove all of the dirt and grime.
Wipe and buff with a Flitz microfiber cloth. They are the best I've found soo far.
Then apply a thin coat of Renaissance Wax the same way you did the Flitz and buff to a shine with a 2nd Flitz microfiber cloth.
I use a toothbrush to clean the grips and a cotton swab to clean where the grip screws go.
I then apply a coat of Renaissance Wax to the grips also.
That should clean it up real nice.