To join the Companions, you must first head to the building Jorrvoskr in Whiterun. Head into the building and go down into the living quarters. In one of the rooms in the basement will be a couple of men sitting at a table and talking. You want to talk to the older of the two, Kodlak. He will tell you that you seem to have “fire in your spirit.” He will tell Vilkas to take you outside and see what you’ve got.
Follow Vilkas outside and when he tells you to take a few swings at him, oblige. After you connect a few times, he will tell you that it was enough. He’ll then ask you to take his sword up to the Skyforge to be sharpened. This begins Taking Up Arms.