I started with a Dark Elf, got a bow and then traded for a better elvin bow and other bows I could find and then simply only ever used the bow unless im about to die to kill pretty much everyhting to get my level up in archery. Simply stick with the weapon and use it the most.
Spoiler Another great tip is go to riverwood (correction) and do a small job for Faendal so can become your follower, he trains in archery. Do the job ask him to follow you, get some money pay him to teach you archery, then ask him to carry something and take the money back, rinse & repeat for archery training.
Spoiler There is also a Khijat roaming with the caravans outside cities named Kyhala ... he can train you sneak skill, you have to do a job for him too and he can also be a follower.
That way you can have two followers who 1. Khijat like you to train sneak and fight with you, 2. A Archery trainer who can be a follower whos great to fight with you equiping him with decent bows like yourself.
Also I joined thieves guild at riften straight away to get by sneak and thief skills up, this combined with archery training has got me a very enjoyable character in this area now.