How do you go about getting a passport from a country other than you birth country?

You can only get a passport from a country of which you are a citizen. It is, of course, possible to be a citizen of a country you weren't born it - there's citizenship by descent, citizenship by naturalisation - but unless you are, or become, a citizen there is no possiblity of getting a passport.

You can get a passport from any country of citizenship, no other country.

It has nothing to do with your birth country. Let's say you were born in Vietnam, so Vietnam was your birth country, you would not be able to get a Vietnamese passport if you are not a Vietnamese citizen.

You don't. You have to be a citizen of a country in order to get a passport.

Dual citizenship is possible, and in that case you could have passports from both countries.

I believe you need to be a citizen of that country before they could issue you their own passport.