Yes, you may do one chapter at a time for a display site, it can help build an audience for your writing.
Here's the
but:
HAVE THE
WHOLE BOOK FINISHED, WORK-SHOPPED, TWEAKED, REWRITTEN, POLISHED, AND EDITED
FIRST.
The reasons are many, but the main one is:
Reality/life interrupts your writing schedule. You may never finish the book because of some catastrophe, you hit a block, the idea fizzles and dies after a strong start, or you grow bored with it. Your readers get impatient, wondering what happened, then eventually figure things out and leave forever. You don't need that.
Posting what is essentially a first draft is a bad idea. You want the best of your best work out there, so avoid impatience, take the time to do it right. Go through the whole writing/editing process the same as you would if putting the whole book up on Kindle.
Most of us want feedback on our darlings 99% of the time. There's no better egoboo than having a pack of readers breathlessly signing on to read your words -- which may never happen if all that's up there is comparable to a plate of raw chicken.
For the sake of your career, take the time to cook it. I can suggest pulling back from all other distractions, focus on the one book and get the first draft done. It takes effort, but it's doable and you feel awesome when it's finished!
While it's in the hands of beta readers, start messing around on the next book. If it's a sequel, issues may surface that require making changes in the first book. If it's a wholly different story, you'll be glad of the break and chance to flex different writing muscles.
What you won't have is the pressure of a chapter deadline looming on that first book.
Dickens has been mentioned as an example, and I'll echo that it was a different time. He got paid a penny a word (one reason his books were so danged long), and if he didn't produce X-number of words, he didn't eat. It was no egoboo for him, but a matter of survival. He was also in competition with dozens of other chapter-a-week novelists in a very small market.
Be aware that the physical act of writing required large sheets of foolscap paper with no lines, a metal pen, and a bottle of ink to dip it in every few words. Trust me, he'd have killed to have our kind of tech equipment -- or even an old Bic!
Ultimately, your name goes on the finished (fully finished!) work. Make sure it is worth your readers' time. If it is, they WILL be back for more!