UPDATE 10/14/2014 : Many are reporting VPN are necessary to purchase books. I've used StreamVia VPN for two years without a problem and recommend them.

For many living abroad, buying Japanese books is a costly and inconvenient task. There are only a few options, and most are pricey import shops. Luckily Amazon JP has a few loopholes to let those outside Japan do this. In addition to that, you can link your foreign account to enable both countries' purchases on the same Kindle (WARNING: read my post about the positives and negatives of linking accounts HERE before trying this.)
What you'll need:
- Any Kindle Paperwhite, iPad, iPhone or Android Device.
- A credit card
- A Japanese Mail Address (fake or real does not matter)
UPDATE 02/04/2015: Free Japanese Mailing Address
OPAS is a mail and package forwarding service that will give you both a US and Japanese shipping address for FREE.
Well, not normally free, but so long as you sign up via this referral link, the $25 signup charge is FREE. You pay ONLY when you want to ship a package. Great, right?
Now you may be asking me, "how does a fake address work for me?" Well, in two ways:
1. Foreign cards aren't checked against their address in Japan. Only expiration date is run through processing systems. Vice versa however does not work as Japanese credit card companies are more stringent due to complaints in the past from Japanese who have traveled abroad and been defrauded.
2. Amazon allows foreign credit cards to be used. I have both a Japanese and US credit card I use regularly depending on the exchange rate. Amazon JP lets me register my US card and purchase freely.
Step 1.
Create an account on Amazon.co.jp. If you don't read Japanese well that's OK. Click the link for English at the top of the screen.

Step 2.
Register your account. Where it says 'Name Pronunciation' just enter the same romaji name again.

Step 3.
Login to this Amazon JP account via your Kindle iOS/Android app, or Kindle Paperwhite. This registers it to your account. Some of you may also Need to login to your US Amazon account, go to 'My Kindle' and select Japan as your 'Country of Residence'. I think it's dependent upon location as I didn't have to.
Step 4.
Browse Kindle Books

Step 5.
Test your device with a free book. 学問のおすすめ is free. Send it to the device you registered by logging in (Step 3) and try it out.
Step 6.
Find a book to buy. If you don't have one in mind I recommend a very quick and easy book called 気まぐれロボット by 星新一. It's fun book and should provide you with a great entry into reading non-Manga.
Step 7.
The registration part. Ok, here's the part where you may have to fib a bit if you don't live in Japan. You pick your address, any address will do and use it for both the billing AND shipping address. I tested this by using the address of the school I went to in Osaka for my exchange program. It worked. No physical product is shipped there so you don't have to worry about them finding out.
Step 8.
Now payment. Non-Japanese credit cards are accepted. Enter your info and it should work even without a correct billing address. If it doesn't, make sure to input at the very least your correct name and phone number, sometimes zipcode for certain cards. Certain US credit card companies require verification against zipcode.
That's it! You should be able to order and read Japanese books on your Kindle app or Paperwhite. A small disclaimer: I'm not responsible if Amazon finds this out and deletes your account, if a giant bear claws you to death or any other action or effect resulting from you reading this post.
Please do let me know if this works for you in the comments! I've tested it using a US proxy and US credit card, on the Kindle app for iOS (also using a proxy for testing purposes). It worked and should in theory work for anyone else.
Got questions? Didn't work? Set something on fire? Comment below and I or someone else may be able to help you.