The first character that comes to mind (and probably the most "postmodern" of all literary heroes) is Slothrop from Thomas Pynchon's Gravity's Rainbow. This is a touchstone novel in the postmodern literary movement and Slothrop demonstrates many traits of this artistic bent.
He is challenged to navigate an incredible series of influences, from jazz to mathematics, as he attempts heroic deeds. However, he falls out of the novel when there are hundreds of pages of story still remaining.
By the end of thenovel, Slothrop's identity seems to have disintegrated altogether, and it is unclear what becomes of him.
This is quite postmodern, indicative of the author's interest in broken stories, in challenging arbitrary narrative conventions, and in commenting on the difficulty of fusing disparate cultural influences.
This novel may be too much of a stretch for your project as it is quite difficult and also nearly a thousand pages.
Other figures you might explore from postmodern literature are Professory Jack Gladney from Don DeLillo's White Noiseand Eugene Debs Hartke from Kurt Vonnegut's Hocus Pocus. Each of these characters is expressive of a commentary on problems of conscience growing from the marriage of consumerism and morality in today's world.
Perhaps these characters and these texts fall outside the scope of strictly "popular" texts and films you are currently focused on. Examples of more pop-oriented, postmodern literary heroes might include the protagonists from The Hunger Games and the works of Steig Larson.