In order to discuss the character of Stephen Dedalus in Joyce's Ulysses, one must first examine Stephen's role in A Portrait of the Artist as a Young Man--the novel in which Joyce first introduced him. In Portrait, readers are immersed into a series of seemingly random episodes comprised of various experiences in Stephen's life. Through these episodes, we see Stephen as a deeply pensive character who questions virtually every aspect of his own existence--from his familial relationships to his religious beliefs and education.
Of particular significance is Stephen Dedalus's name. Primarily, Saint Stephen is believed to have been the first Christian martyr who was put to death after being accused of speaking against God and Moses. As Christianity--specifically Roman Catholicism--is central in both Portrait and Ulysses, Stephen's first name is not an accident. Further, the name Dedalus is remniscent of the Greek architect Daedalus, who, according tp mythology, created wax wings to escape the island of Crete. (Daedalus's son, Icarus, flew too close to the sun, despite his father's warning; as a result, the wax in his wings melted and he crashed.) As Daedalus was a thinker and innovator, so too is Stephen.
As Ulysses opens, Stephen is struggling to cope with his mother's recent death. In the presence of his jovial roommate, Buck Mulligan, Stephen is unable to emerge from his somber mood--a mood that follows him for most of the novel.
Though Ulysses is an intricately-constructed work of literature whose complexity is difficult to discuss in short, Stephen Dedalus can be described as an intellectual scholar whose troubled relationship with his family and the Catholic church drive him to seek for his own purpose. Ultimately, as he is relatively alienated from his own father, Stephen develops a brief, though meaningful, relationship with Leopold Bloom--another of the novel's main characters--who is also seeking to fill the void of the infant son he lost years before.