Skyrim Fast Leveling Archery
In the first town you get to after the initial dungeon (it is also the place that you are directed to by the first quest) find Faendal (he is usually walking around the lumberjack or can be found in his house). Complete his quest making sure that you favorite him and not the other guy. Then go back to him and ask him to train you. Purchase training until you run out of money. Then select that you want to trade some things with him and take back your gold.
It is best to do this as early as possible because then you can level up with his training and keep on doing it to the max level of 60 that Faendal can train you. Also complete all 5 times that you can train so that you gain maximum level to keep the loop going for as long as possible. In the video you can see that I keep on buying training even though I have actually leveled up.
There is a trick to leveling archery very quickly up to level 60 once you reach the first town in Skyrim. It is best explained if you view the video below: