I've seen posts for this sort of thing around, but things changed since the article I looked at, so here is the procedure I went through for getting a Japanese bank account:
When I went to register at the city office, I had my address written on the back of my residency card, and stamped. Apparently, this was sufficient proof of address (they sounded a little unsure, so where possible, I'd advise having an insurance card, but even so, I managed to get an account this way).
You'll then have to fill in the forms yourself, in Japanese. Apparently after some incidents or other it is now necessary that the person setting up the account actually do all the writing. Still, if you have brought the proof of address, then you can copy the characters off of that; however, you'll also need to be proficient at writing katakana, both for your name and for the pronunciation of the address.
A handy thing about getting a bank account with the post office is that you can register yourself at your address at the same time as making a bank account. I'm not sure how important this is, as I've sent things addressed to my daughter, and to my knowledge she isn't specifically registered with the post office, and I've never had anything lost in the post...
The other handy thing about a post office account is that they have branches all over the place in Japan, which is great if (like me) you don't know exactly where you'll end up.