A white vinegar and water mixture should clean it, just don't get any on the frame. If it's cloudy or has black spots, the 'silver' has been damaged.
I love the feeling of old mirrors! Can you show picture of your mirror?
I use microfiber and water to clean mirrors. SuperCloth is my favorite, it's similar to the cloth that you receive when you purchase a pair of glasses or sunglasses.
You can clean an old mirror with whatever you would use on a new one. But with any mirror, it is generally better to spray the product on the cleaning cloth or paper towel than directly onto the mirror because it can run down and settle in the frame where it can damage the frame and possibly the backing of the mirror.
If cleaning the mirror doesn't remove the spots and sort of haze you might see, then it may well be that the back of the mirror has been damaged over time and you might not end up with a really good reflective mirror. In that case, use it more as a piece of art than for typical mirror uses. Having an old mirror resilvered is ridiculously expensive and is generally not worth the cost.