A hair dryer is an absoloute NO-NO when it comes to waxing your string. The heat from the dryer is far too high for most bowstrings and cables. You can cause premature creep of your string and significantly shorten the life. Not to mention peep rotation, rest timing issues and cam timing issues.

As mentioned above, the best way is to put the bow in the press, relax the string and cables then work the wax into and between the fibers of the string. Use your fingers or a piece of chamois to rub and warm the wax, making it easier to get in there. You are only trying to warm the wax, not start a fire, so don't go at it like a teenage boy who has found some "alone time"

Swamp Collie has some good advise about the dirt and grit. Prior to waxing the string, I use a piece of serving material or dental floss to wrap around the string/cables then slide the loop down over the string. This scrapes off the majority of the old wax and particulate from the surface, allowing you to get all of the dirt and grit off before they get worked deeper into the string. It also helps prevent wax buildup.