To level up your Archery skill quickly you need to go to Riverwood and seek out an elf called Faendal. You also need 200 Gold.

ou will be required to deliver a note to a person named Camilla. Then go back to Faendal and ask him to follow you.

After that, ask him to train you and when he does, ask him to trade with you. You can go into his inventory and simply take your money back.

This works up to five times each level!