You need to have some creativity while writing a script and in order to become a TV story editor you first need write a few sample scripts of the shows which are already on air. You could choose to write a pilot script of your own show.
You need to have an agent who will help get your script to production studios if he/she feels that the script has enough to grab the attention of the producers.
If you haven’t had an experience of being a story editor, you will have to start from the post of staff writer. Here you need to pay special attention to your work and learn almost everything about this profession. It is imperative that you learn from the writers of the show by attending the writers' room meetings. You will be tested here by overburden of work, but stay strong, do your job professionally as it will polish your writing skills.
If the show’s producers are impressed with your work, you could be asked to come back for the second season, if this happens it is suggested that you try to negotiate your position and convince the producers to hire you as a story editor. And if you are seeking to work on some other show, then have your agent search for a position as a story editor for you. With experience behind you, and with so many shows coming out every season, you will soon be hired as a story editor.