What's the best way to go about teaching English in Japan? The good news is that there are quite a few. Some of them even offer you a free plane ticket!
I'm probably biased here, but I'd say the best English teaching job in Japan is the Japanese Government's JET Program. That's how I first came to Japan. It stands for "Japan Exchange and Teaching" Program, and although you will probably be in schools teaching English for several hours a day, the main aim of the program is to get local communities used to foreign faces. Just imagine if in your town back home no one had ever seen someone of a different race or culture, only on TV. Well that's what most of Japan is like. You're basically paid to be a foreign face!
The JET program brings over several thousand people every year and disperses them all over Japan, from the snowy north of Hokkaido to the tropical south of Okinawa. And no, you usually can not choose where you'll be! Although you can give a preference. The government also pays for your flight over.
It is a great job, but it's very much "every situation is what you make it". If you're independent, can come up with ideas to spend your time and abilities, and have the patience and people skills to open people's eyes but at the same time respect their thoughts and positions, it's the job for you.
Otherwise you may consider a straight teaching job, without any of the "internationalization" responsibilities. If you were wanting to speak Japanese, you'd want to learn from a native Japanese speaker, would not you? And it's the same in Japan, people want to learn English from real native speakers. These "eikaiwa" jobs, as they are known, are plentiful in Japan. As the pay and conditions are not always super wonderful they are sometimes refereed to as "mac jobs", but if you are the sort of person to make the most of any situation it's a great chance to visit an exotic country, meet the people and get paid for it.
The top of the chain is to start your own school. That's probably a little way off for you yet, but it always pays to have a long term goal!
In any case, enjoy your time here. It's a great place to live and teaching English in Japan is a fantastic way to give something back.