According to the website of MOFA (Ministry of Foreign Affairs of Japan), teaching English requires Working Visa and its status of residence is Instructor.
Validity of this visa is from 1 year up to 3 years, according to the site.
Statuses of residence for engaging in specified activities in Japan
(a) Statuses of residence permitting work
http://www.mofa.go.jp/j_info/visit/visa/04.html#b1a
My understanding is to obtain a working visa for working as an English teacher, you'll need
- a university degree (or several years experience as an English teacher)
- a prospective employer as your sponsor
In case of private teaching, your students-to-be would be prospective employers but I'm not sure how much it is valid. Maybe it is easier to get a contract with a Language school as one of English teachers and later switch to private teaching. But this is only my guess, I'm sure there are many members who has the experience in teaching Japan and who will be able to give you more appropriate/precise advice.
Brief information on visa and working in Japan
Entering Japan
Working in Japan
There are a lot of information concerning VISA, which is rather complicated. I'd recommend you to take a look at this page for details.
VISA
Just for your reference.
I wish you all the best for you!