After berating my younger sister for the tripe that she watches on television (keeping up with the kardashians, etc.), I became curious as to this question. It's no doubt there are much more important issues to be talked about today but I wonder if societies of the past had the same problem to a degree, if at all, or had it primarily been the product of a media saturated culture?
Did Romans really give a crap about what color toga Julius wore? Did peasants gossip about Henry VIII's weight?
Well, I'm not sure whether you mean ancient like the classical world, or just "pre-Page Six" as your Henry VIII example suggests. Hopefully some of the classicists will be along to talk about gladiators--the popular ones even had stage names, and posters advertising them--but I can tell you that the European Middle Ages absolutely had celebrities!
I'd call it more of a "fama culture" (fama is public fame or reputation) than straight-up celebrity, though--the Middle Ages really put it to use, in part because so much communication and social operation was based on oral culture and personal relationships.
For example, fama had legal status. Good fama and bad fama (infama) were legally admissible evidence in court cases!
Also, a big way that princes communicated with their advisors and ambassadors was letters. We have massive letter collections from key players like Hans von Planitz (lawyer and advisor to Frederick the Wise--you know, the dude who basically controlled who got chosen Holy Roman Emperor, the dude who stood up to the pope and that emperor and sheltered Martin Luther) that are about eighty percent "Here is what is happening at the Reichstag, here is how the Reformation is tearing apart the Hungarian royal family" and twenty percent, "Everyone knows the duchess of Marburg is the best dancer at court." In other words, making sure the elector was able to function in the social environment of politics as well as the political one.
As far as particular celebrities for the Middle Ages, you'd do very well to look to "living saints"--this is the mode that Anna Laminit was working in when she made herself the (very wealthy) pride and joy of 1500 Augsburg, albeit as a con artist leeching money for herself rather than someone actually trying to be pious. In fifteenth century France, Joan of Arc spurred imitators after her death who wined and dined with people who threw open their homes and hearts to them because the people thought they were dealing with the real Joan. In twelfth century Germany, monks and bishops wrote to the nun Hildegard of Bingen for advice and spiritual consolation, either because they'd heard about her visions of God or because they'd heard her music being performed in Paris.
I don't know as much about outside Europe. However, in much of the Islamic world, Sufi saints absolutely attracted admirers and visitors similarly to European ones. The Kashmiri poet-mystic Lalla or Lal Ded (d. 1350s) was so popular that she was claimed by Hindus and Muslims, who each built their own shrines to her.
Caring about people's weight...the example I have for that is later, it's Elisabeth-Charlotte von der Pfalz, Duchesse d'Orleans at the court of Louis XIV, writing the most insanely gossipy letters to the nobility of Europe and her kids. She has quite a bit to say about other people's weight...and her own...and the prince of France throwing his wig at people when he loses chess games.
And again, the important thing to remember is--this isn't "trivial" stuff. (Maybe in 20 years, there will be a /r/AskHistorians thread about how celebrity culture in 2015 had Deep Meaning, too...) Knowing these little details actually gave people power then, gave them the ability to operate in the political-social enviroment that still functioned so much on relationships and reputation.
There was definitely a kind of celebrity culture in the ancient world. Succesful athletes at the Olympic games could be immortalized on pottery, frescos and votive offerings. Poets and playwrights who won the various competitions and festivals in Ancient Greece would see their works distributed far and wide and could find themselves hired to write poems for important occassions or for the glory of some wealthy patron.
A professor I had at college told us about herms. They were statues set up at property boundries and they had a bust of Hermes (bearded, indicating their archaic origins) on top and an erect phallus at where you would expect it. By the time of Christ they had largely lost their religious functions and people would use busts of philosophers, politicians, writers, orators, etc they admired instead of Hermes.
And of course, Herostratus burned down the Temple of Artemis at Ephesus (one of the Seven Wonders) in a bid to become famous.
/u/sunagainstgold mentioned the Duchess of Orleans in the time of Louis XIV, but the Duke of Saint-Simon left three volumes of gossipy memoirs.