"Let me call myself, for the present, William Wilson. That is not my real name. That name has already been the cause of the horror- of the anger of my family (Poe para.1)."
"The words were venom in my ears; and when, upon the day of my arrival a second William Wilson came also to the academy (Poe para.10)."
In the beginning, the narrator doesn't state his real name and prefers to be named as William Wilson, because he had a troubled past and was wronged.
"..as i stepped up to it in extremity of terror, mine own image, but with features all pale and dabbled in blood (Poe para.31)..
"In me didst thou exist--and, in my death, see by this image, which is thine own, how utterly thou hast murdered thyself (Poe para.47)."
William recites a moment in his childhood that when he went to school, he was the superior one until he met his doppelganger (twin). Thus begins the conflict in the story.
Throughout the years, William Wilson goes through a series of vice or evil. His twin is essentially there to follow him in whatever he does to prevent him from doing any mischief. When William engages in gambling with his fellow noblemen, his twin basically snitches on him for cheating.
After stabbing his Twin, William looks at the reflection of the mirror of himself pale and splattered with blood. Revealing that he has killed his true self.
When William realized that he has basically killed himself, he has no will to live and is dying inside.
William arrives at a masquerade ball and tries to have an affair with a married woman. His plan fell short again when he is met with his double wearing the same outfit. Outraged, he quickly kills him in a private room.