Tenhou offers multiple ways to buy an ID (3 ways?). These screenshots show the process of buying an ID via credit card.

It's pretty straightforward. All you might want is an autotranslator handy in case you want to know what something says. The text translates pretty well.

1. Put in the first half of your ID here.
   
2. You can just skip all of this and push the button, unless you happen to know Japanese.
   
   
3. Pick how many days you want and the price. You can use xe's currency converter to see how much it will be in your currency.
   
   
4. Fill in your details and you're more or less done. Double check them and then press the order button.
   
5. Success! The date in bold is the date your ID expires as a premium ID (it will still work afterwards, though).
   

That's all there is to it.