You're right not to try applying cleaning products without fully knowing what you're doing. If you have a violin repair person or luthier in your area, I recommend you take it straight to them. They can give it a full beauty treatment, removing any layers of dirt and polishing as appropriate for the instrument. But more importantly, they can assess any repair that the violin surely needs, and even appraise its value. I would absolutely try to make this visit before trying to play or sell the instrument; even if there were no such expert near me I would travel if possible.
If you absolutely can't get it to a professional, I might wipe it gently with a soft (like microfiber) cloth and no cleaning product at all. (Even then, as Lazy's comment below mentions, you must wipe very gently to avoid scratching. If in doubt, leave it alone.) It also can't hurt to "air it out"; you can get it out of the musty case, if you have a nice secure place to store it where it's protected and the room is not too hot or dry. If you plan to keep the instrument, you can also start shopping for a nice new case; modern cases with suspension pads or carbon-fiber bodies can protect it better.
NOTE: Although the question asks about doing an intensive cleaning on a very dirty violin, for the benefit of other users: how do you do regular daily cleaning on an instrument that's already in good playing shape? DO keep a clean, dry, soft cloth in your case; when you put the instrument away, wipe rosin dust gently from the entire belly including under the fingerboard and tailpiece. Wipe rosin dust from the bowing area of the strings and wipe sweat and oil from the fingering portion (it might be smart to use a second and third cloth for these, so that your body cloth stays clean longer, and you don't get the oil/sweat on the bowing area). Wipe rosin dust from the stick of the bow as well. That's it. Don't use any kind of product or solvent on any part of the instrument, no matter how it's labeled or marketed, no matter what Yehudi Menuhin used to do.