Who is related to both Lady Capulet and the Prince?

Prince Escalus represents something of a voice of judgment within the play. He strives to keep the peace, and at the end of the play, he tells the Capulets and Montagues that their blood feud has not only resulted in the deaths of their children, but also of two of his own kinsmen—namely, Mercutio, whom he has earlier identified as "my blood," and Count Paris, who is another of his relatives. Prince Escalus does not anywhere in the play identify anyone else as being related to him.
I don't think either Paris or Mercutio is related to Lady Capulet or the Capulets in general. Paris wants to marry Juliet and is not part of the Capulet family. Mercutio is related to both the Prince and Paris, but his loyalties lie not with the Capulets but with the Montague tribe to which his friend Romeo belongs.
He does get invited to the Capulet ball, which is perhaps slightly strange if there isn't any relationship between him and them, so presumably they do know him. You could argue that he may know Lady Capulet through her maiden family—although this is entirely speculation, as we don't know much about Lady Capulet's life or familial associations before she was married to her husband as a young teenager.

Mercutio, Romeo's best friend, is related, by blood, to Prince Escalus. We learn of their connection after the fight in which Tybalt challenges Romeo. Romeo refuses to fight (because he's literally just married Tybalt's cousin, Juliet), so Mercutio fights Tybalt because he believes Romeo dishonors himself by refusing the challenge. Tybalt then kills Mercutio when Romeo comes between them during the fight, and then, finally, Romeo kills Tybalt. After Romeo has fled the scene, the Prince arrives and Benvolio explains what happened. Upon hearing the story, the Prince says, 

I have an interest in your hearts' proceeding.My blood for your rude brawls doth lie a-bleeding.  (3.1.150-151)

In so saying, the Prince acknowledges that Mercutio was his relative. Further, Mercutio's family had received an invitation to the party at the Capulets' house, and so this indicates, perhaps, some kind of relationship, even if not familial. Otherwise, I'm not sure who might be related to both the Prince and Lady Capulet. The Count Paris is also related to the Prince, but there is not evidence to suggest a relationship to Lady Capulet.