My name is Peter.

I said, "Good morning. Hi, my name is Peter."

I'm from Washington, DC, in the USA.
I said, "I'm from the United States, from Washington, DC."
The USA is correct but I think Washington, DC is a better answer because it is more specific.

I teach English.
I said, "Where I have been an English teacher. I was also an English teacher in Japan. Now I'm teaching English in Vietnam."

"What do you do" Is a question about work. I do live in Vietnam, but it is not what I do for work. I was an English teacher before and I still am so the answer should be in the present tense. I do come from Washington, DC but that is not my job. If I were an airplane pilot... maybe.

I didn't really answer the question. Sorry!
I said teaching English helps me learn more and more about the language.
I finished with, "I hope to be fluent in English one day too."

I should just say that I am fluent. I am a well educated native speaker of English. I speak English fluently, but I want us to remember that fluently does not mean perefectly. We all make mistakes and we all have a lot to learn.

I want to learn Vietnamese and Japanese.
In my answer I did not use the word LEARN or STUDY. I repeated FLUENT. I will have to learn a lot more Vietnamese and Japanese to become fluent.

I said, "And I hope to be fluent in Vietnamese as well ... and Japanese."

I did answer the question, but I answered the question in a different part of my introduction. When I talked about being fluent I said, "I learn so much more about the language by working with people like you." And I said that being in the group will help me become more and more fluent in English.