Regarding cleaning marble, you should take care not to use any agressive materials, particularly not anything with bleach in it. Marble is a slightly fragile stone in that atmospheric pollutants tend to attack it and essential salts get leeched out - acid rain is really deadly for this stone. In fact, for outdoor uses in Britain, marble would be well down the list of my preferences due to the amount of dampness above freezing point, especially through autumn, winter and spring when algae thrive. It is quite possible that the marks you see on the stone are in fact both living and dead algae. Be prepared to have to accept that their removal is not going to be successful. The best way to try is with a mild solution of washing up liquid and at most using a plastic scourer to gently rub at the worst bits that won't come off by rubbing firmly with a cloth.
as for sticking the letters on, a solvent based adhesive is probably best, or else an epoxy. The latter is a bit more difficult to control, but I am thinking in terms of No Nails or a similar construction adhesive. Be very careful not to use too much adhesive because the excess will be tricky to remove. I would suggest you use masking tape to put on the stone to give just the right footprint for each letter in turn. Have a small knife handy to immediately remove any excess adhesive and also something to wipe the adhesive off on. This work requires care and patience.