Many moons ago it was my profession as a Monumental Mason, ideally it should be cleaned by a monumental mason for best results.
firstly are you absolutely sure it is marble? (usually white often with grey in it)
The lettering is what you need to be very careful of, this is made from inlaid lead! "DO NOT USE ANY BLEACH" it will turn the lettering a very dark brown and may stain the marble. I would suggest washing it thoroughly using a plastic scourer until you have got the worst of the grime of and finish off using a piece of "wet&dry" carborundum paper, about 220 grit, available in any DIY shop and use it with plenty of water and rub it all over until clean. This is not an easy 5 minute job, depending on the design of the headstone and how dirty it is. If I had a good picture of it I could probably give more precise and detailed advice. You may see the lettering turn a bright silver as the lead becomes clean again, this will fade back to a darker colour in a few days, you can use the "wet&dry" carborundum paper all over the lettering with no problems. DO NOT USE THIS METHOD UNLESS YOU ARE SURE THAT IT IS MARBLE! Good luck!