So you want to be a "Companion"? Well, there’s two ways of going about it.
Note: Of particular note here is the locked display cases in the Jorrvaskr Living Quarters. There’s a NICE bow here (especially for those just starting out). You can also find an Archery Skillbook here called "The Markmanship Lesson" in one of the rooms. There is another Two-Handed Skillbook called "Song of Hrormir" in a different room.
Once you find Kodlak tell him you’d like to join the Companions. He will look you over and say you may have what it takes. He’ll then ask how you are in battle. Answer however you want and after a brief interruption by Vilkas, he will tell Vilkas to go test you out. This ends the MISCELLANEOUS QUEST that brought you here and starts a PROPER quest: " Take Up Arms ".