So you want to be a "Companion"? Well, there’s two ways of going about it.
It's very likely that as you first approached the town of Whiterun you encountered a group of people fighting a giant/troll/animal out in a field. Hell, chances are you tried to help! Once the beast is slain, talk with them and you will receive a **Miscellaneous Quest** called **"Speak with the leader of the Companions".**
Or you can visit their place of rest known as Jorrvaskr in **Whiterun** (it's to the right of **Dragonhelm** on your map) and speak to someone inside about joining. They'll tell you to speak to Kodlak Whitemane, who can be found downstairs in the living quarters.
This is the best **** option as you're not relying on a random encounter to trigger.
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 ".