There are multiple tasks to do before you can become a member of the Circle, some mundane (and not counting as guild quests, but necessary to progress the questline) and others substantial and making up part of the questline. They are as follows:
Hired Muscle – Intimidation is the name of the game here. Short and easy quest.
Proving Honour – You’ll be asked to retrieve part of a blade here.
Rescue Mission – The game gives you a choice of three missions to pick from, from the people wandering around Jorrvaskr: Aela the Huntress, Skjor & Vilkas (we chose Vilkas, whose job was to save a kidnapped citizen in “Rescue Mission.”) Whether that changes, who knows.
The Silver Hand – After returning to Whiterun and to the Jorrvaskr, having completed the mission, you’ll have to meet Skjor at night round the back of the building, before the initiation begins. Note: you don’t have to complete this mission to get the achievement, you get it after the ritual… plus, you need to be quite powerful as the end boss-like person in this quest is a little powerful (talking from the perspective of a level 20), but to get the final Companions achievement, you have to soldier on, complete this and do a number of other missions.