I found this to be one of the easiest skills to increase naturally. There are a lot of dungeons and caves in Skyrim and the main story progression will take you through a lot of them. There will be a lot of caves where you find sleeping Drauger. If you sneak around these guys and attack them with your bow, you will increase your sneak skill and bow skill. Since it is a sneak attack you may also kill the enemy in one shot.
In general the best way to raise the sneak skill is to attack an enemy with a dagger from behind while sneaking. If you can do this while remaining unnoticed, you get a massive damage bonus and a large amount of points toward the sneak skill. You can spam this on the graybeards, sneak behind them and keep attacking. When they stand up to attack you, stop and they will kneel back down, keep doing this over and over.
There is also an even easier way to increase your sneak skill. You can find a city guard who remains standing in the same location all the time. Typically someone standing on top of a tower or something. Make sure they dont notice you. Usually you can walk against a wall and get sort of stuck so that your character keeps walking, but you are not actually moving anywhere. This counts as sneaking and will slowly raise your skill. If you tape stick or button down to move, you can go off for some coffee while your sneak skill is levelled up.