How to Become an Animation Writer
Becoming an animation writer is a difficult and rewarding task. There truly isn't that much difference between becoming any kind fiction of writer and becoming an animation writer. There is the old saying, "You will write 500 pages of drivel before you write one page you can use." It takes a lot of persistence, a decent education and a little luck.
Learn how to write a script
Animation is written in the same way as live action television or film is written. Animation follows the same technical rules, in terms of format, and artistic rules, in terms of narrative structure and character arc. A great place to learn this is a film school, or if that's not something you're wanting to invest in, there are many books on the subject. One in particular, "The Writer's Journey", written by a former Disney development executive, is used by many animation specific production companies.
Consider internships, assistant positions and PA gigs
These are great doors into the industry. You can learn the ropes while working. Finding these positions can be tricky, but the payoff is more than worth it. An assistant's position is one of the few entertainment jobs that acts like an apprenticeship. It can lead directly to the next rung on the working writer's ladder.
Keep writing
It cannot be stressed enough, you need to keep writing. Writing skills are like a muscle that get stronger the more you use them and atrophy when not exercised. Write all sorts of stories, and be sure to rewrite them as well. Polish and make them presentable. You're going to want to have at least one strong "calling card" script. A script which you can hand out as an example of your absolute best writing.
Be persistent
It can take at least 6 years to have any success in the entertainment industry. It is not always the most talented who make it as writers, but the most persistent. Those willing to work, get work.