Though, as I think about it, "the" writing process implies that there is only one correct way in which a person can write something. Maybe more accurately, I'm going to talk about my writing process.
I'm not a big fan of outlines or scene breakdowns. I find that they limit the creative possibilities. I do have a general story idea for each episode, but it's much more exciting to begin writing without any restrictions on scene.
As I write, it's almost as if I am reading along, because I don't know what is going to happen next. Sometimes I feel like the characters are writing the story themselves. This all might sound a little crazy, especially if you're not a writer, but it makes sense to me.
I think that if you know your characters, you know how they behave, and when you put them in different situations, it's easy to step into their shoes, and write what they'll do next. The choices that they make can open up new possibilities, and I'm often very surprised by where the story goes.
After my first draft is written (outline free) I go back and re-focus the plot. Often, I'll have numerous plot-points and side-stories that need to be trimmed, or cut entirely, because they don't necessarily fit in with the larger story that I decided to tell.
I always save what I cut, because sometimes there are great scenes that can be used in future episodes. I have a whole folder filled with "b-stories" that might one day be inserted into a new script.
When I'm done re-focusing the story, I go through one more revision, where I clean up the entire script. After that, I have a fresh episode, ready to be filmed.
So there it is... My writing process!