How do you use the Japanese character keyboard?
I think you don't actually need it for the course. They enabled the tapping system on browser just so they could release the so required Japanese course.
Now, if you still want to have a Japanese keyboard for other purposes, you can download Google IME Japanese, which is the one I use. The link is here.
I'm not sure what you mean by "taping system on browser", but recently, the web version of duolingo has been requiring typed Japanese input. As for the original poster, you will need to activate an optional Japanese keyboard which you will likely find under your computer's keyboard settings.