Harrison Bergeron didn't kill himself literally, but he did instrument his own killing by standing up for what he believed in. Standing up for one's rights never goes without some consequence, but in Harrison's case it led to his demise. Harrison felt what they were doing to people was wrong. He felt everyone was entitled to his own individuality and that no one should be punished because he was smarter or better looking than someone else. Being different is what the world is all about and we all should embrace the differences and rejoice for what we have to share with others. Harrison tried to initiate a change in his world and was willing to sacrifice his life for it along with his Empress, the ballerina. This could have been a new start for a new world, but Diana Moom Glampers put a damper on his quest for individuality when she shot him and his Empress dead. This showed that there was a fear for change and that individuality was not admired in a person. In this world monotony and stagnation were admired and there would always be a unrealistic world of controlled robots. People are always threatened by change.
Source: The Language of Literature Book by McDougal Littell
He didn't--not in the short story, anyway. He was shot by Diana Moon Glampers, the handicapper general, after declaring himself emperor.
In the film version, Harrison kills himself because he knows that his death--shooting himself in the head on live television--is the only way to keep his message to the masses alive. He would rather die in this way that makes people sit up and take notice, and maybe take off thier handicaps, than exist in a world where people are taught to accept simply being controlled. He has tasted the sweetness of REAL art and REAL beauty, and he wants other to share that same experience.
He knows that if he doesn't make some drastic move, then everything will remain the same, and no one, except possibly the secret society that runs the country behind the scenes, will understand the importance of individuality and excellence. He sees that as a tragedy.
And his plan works--if you notice, it is his son who later watches the videos of Harrison sharing art with the world, and his son and friend take of thier handicapping headsets.
Actually, General Diana Moon Glampers kills Harrison and the ballerina instantly with a double-blast of her shotgun.