Eddie Haskell from Leave it to Beaver was a pretty good example of a perpetual bully.
Not a TV show, but a movie, The Diary of a Wimpy Kid, show’s the older brother of Greg (the wimpy kid). The bully brother is named Rodrick.
And Glee, although it is for slightly older kids (I think a lot of 4th graders watch it) has a number of bullies. Most of the football team (the ones who aren’t in glee club) regularly make snide comments and throw slushies at the members of glee club. And one guy on the football team, Dave Karofsky, made all sorts of bullying comments and physical altercations and threats towards Kurt (played by Chris Colfer). Curt is an openly gay member of the glee club, and Dave Karofsky was a very mean, closeted gay football character.
A recent made for TV movie, called “Cyberbully” starring Emily Osment (who plays Miley Cyrus best friend on Hannah Montana) was all about mean girls and boys posting inflammatory remarks on Facebook to the point that Osment’s character attempts suicide.
The older brother, Reese, on Malcolm in the middle is a pretty bad bully.
And of course, in the movie Harry Potter, Draco Malfoy is a very bad bully.