While I can't speak to the pace of posting chapters, I can speak to the pace of writing. I know you're throwing "week" up there just as an example, but I would definitely give yourself more time to spend on each chapter, since you have to get the idea spewed forth onto the screen and then edit it to a nice, presentable state. A week may seem like a lot of time, but then you sit down to work and suddenly seven days are gone like that, you're six chapters in, and you have no relief in sight. The severity of that feeling is dependent on how long you make each chapter, of course, since shorter chapters will require less time, but a hobby can start to feel like an obligation very quickly if you don't leave yourself time to breathe.
Consistency is key for maintaining and growing an audience. A posting schedule with consistency, like the second Thursday of every month, or every other Tuesday, will give people a sense of regularity that will ensure they return on their own to check out the latest installment of your tale, but it requires you be faithful to that schedule. This is where the pocket full of chapters comes in handy. With a few pre-written and pre-edited chapters standing by before you ever post the first ensures you can uphold the schedule without panic or stress, and gives you the freedom to take a couple extra days off when you need them. Three strikes me as a good starter if you're eager to get to writing and posting. Personally, I'm both lazy and gung-ho, so I'd rather have the whole thing finished and edited to perfection before I start posting, just so there's NO chance I could ever fall behind, but that's just me and I know I'm terrible at keeping up with posting regularly if I don't have material already prepared.
Just to give you an idea of what you're looking at once you start putting words to the screen, NaNoWriMo, or National Novel Writing Month, is a challenge for writers to complete a novel's first draft in thirty days. The minimum requirement is 50,000 words. That's about 1,667 words a day, or 2,500 if you take two days off a week. Some people struggle to write 1000 words in an entire day while others can breeze right on through it. Those who breeze often do so because they spend months in advance planning, so the writing is more like filling in the space between fixed plot points, rather than predominantly forming it as they go. So generally speaking, while every writer's process is different, having a clear map to follow will help you get there much faster, and with much less wandering.
Many writers use programs such as yWriter and Scrivener to organize their chapters, plots, and characters. I can only tell you about yWriter from personal experience, but both have a lot of the same functions. yWriter is more stripped down while Scrivener has many more bells and whistles, but each allows you to enter a chapter in its own window, where it tracks word count, characters involved, scenes involved, locations, objects, themes, goals, tension, just about anything you'd want to track, and a million other things you might not even realize you wanted to track. It consolidates everything into one location (yWriter itself) so you can flit back and forth between all your pieces without having every word document and spread sheet open at once. The only real obstacle is its lack of spell-check, but you could always compose the chapter in a word processor and transfer the text to yWriter (as I do) when you're done for the day.
Whatever you choose to do, remember not to let any method, format, or schedule you choose to use sap the fun out of writing your story. Especially if it's your first of this length. Writing a story or book, fan fiction or not, can be a lot like flipping a house. You look at it and think "This won't be so bad," and you research and plan and get your stuff together with the confidence that you've given yourself enough to handle anything this project can throw at you, and then you find out the foundation is cracked, there's water damage in the basement and first level walls, termite damage on the East side, and the whole thing needs to be re-plumbed and rewired. Suddenly this project you thought you'd prepared yourself for is consuming you, making demands you never thought to account for, and you're left with the choice of abandoning it, or seeing it through somehow. Don't box yourself into a corner in the rush of excitement that can come with wanting to share your work. Really think about a schedule that feels comfortable and natural to you, that your current RL schedule can support with wiggle room for any unforeseen bumps that might arise along the way.
And keep HAVING FUN.
