If I am Filipino, what are my chances of getting hired as an English Teacher in Japan?

If you want to find a placement with a reputable teaching agency then the answer is yes.
Howerver you can go it alone as there are opportunities for private teachers in Japan. Your english has to be very good to get a post. If you have a recognised degree in a major subject a lot of agencies will let you teach without TESOL

Not likely. They prefer native speakers. You can work on the JET programme in Japan as a native speaker if you have a degree. But it is more likely you'd get a job arranged privately if you have a TESOL (e.g CELTA/Trinity Certificate)

Check out - eslcafe.com - It's a huge website with many forums and there is a Japan forum where you can ask your question. It also has a huge Job board for almost any country in the world. I think you'll like it.