A Clique is your circle of friends from your fellow students.
You can add any number of friends from any of the seven Colleges, but you will need to balance their mutual Relationships as well as you own to everyone of them.
Adding someone to your Clique
For adding a Student to your Clique, you will need to use Action:Befriend on them, winning a contested Roll (Charm/Befriend. This will be a lot easier if you have a high positive Relationship to the person in question, so you might want to expand this first to, say 4, depending on your skill and Charm.
If you are already member of a Clique, you cannot befriend someone from a different Clique. You possibly have to break him away from his Clique by lowering his relationship with the other members greatly.
Founding or Entering a Clique
If you first befriend a Student who already is a member of a Clique, you will become a member of his or her Clique.
If you befriend a Student, who is not a member of a Clique to begin with, you found your own Clique and can baptise it as you choose. [Now (as of DLC ?) you must enter a Clique name before even commencing game play; also, your character is always 'in your Clique', i.e. you may only recruit others into it, and you can no longer be recruited into any other Clique.]
Every Student apart from your character has some special access, behavior, info, status and/or talent which in game terms translates into a Clique Ability.
This might be an Ability, such as Expanding Skills of yours, boni to Abilities or Skills, or a decreased Chance of Discovery, for example, which may be passive and stay with you all the time, or you can actively use time and again.
If Relationships between individual members of your Clique get too low, they may split up, some founding one or more new Clique(s) or leaving just so, leaving you with the choice where to stay. Sometimes a Clique disbands just because.
You may try to split up another Clique (or getting someone to leave) by lowering Relationships between their members by social actions, e.g. by using Demean (Action).
First you will need to bring up your School Survival - Social Skills up to at least two.
First level to add the Befriend (Action) to yourself, second to get Informed about the Befriend Skill, so you can Expand your Relationship to someone and stand a chance using the Befriend Action itself by raising that Parent Skill. Expanding Conversation from that skilltree gives you your first Action to Expand Relationships.
Be aware, that the Gossip (Action) not affect your own Relationship to someone. While it is not a bad Skill to have, it should not be your first priority.
This almost comes naturally, as you will suffer negative Emotions until you do.
When using Action:Befriend, us a suitable location from the menu brought up when you confirm the actions for the day. The Garden Gallery will give you a bonus of 2, The Great Hall will give you +10% Chance of Success and it's automatically explored at the beginning.
A good start for a Clique might be Durand de Thiomines, whom you will automatically share an adventure with on Fool's Day, which if successful will greatly enhance your Relationship. He has a very useful Clique Ability which allows you to Expand 3 random Incantation Subskills once a week.
If you fancy someone else, who has already joined a Clique, get that one first.
Actually, Adventures can be a good way to Expand your Relationship with someone. For to unlock the Advenbture pertaining to him or her, you might have to raise your mutual Relationship to 2 or 3 first.