Which kind of document i have to do after born baby?

chiavetta wrote:

Father foreigner - Mother Vietnamese
Baby born in other country (got double citizenship and passports)

Who has double citizenship?  The mother or the baby?  Do you mean that the baby is a citizen of two countries other than Vietnam or Vietnam and another country?   You are very unclear about this.

chiavetta wrote:

Now
how to register baby on "family book"?

The baby's mother should do this, not you.  Is the mother in the family book of a relative?

chiavetta wrote:

Could register also the father on it?

Probably not.  It is for citizens.  Anyway, why would you want to?  It is a mechanism for control of the population by the government and the Party.  You need to be registered as a temporary resident at the local police station if you are residing in Vietnam.  As your profile says you have been in Vietnam since 2004, I certainly hope you have done this.  As you told us that the baby was born in another country, perhaps that date is inaccurate.

chiavetta wrote:

Where do you have to go to do it?

Each family has it's own book in it's possession.  A copy of the contents is kept at the local Communist Party office (UBND).    If someone is to be added, the head of household must bring the book in to the UBND.   The mother may be in her parents book if they still live in Vietnam, or she may be in the book of another relative.

chiavetta wrote:

How long is the procedure?
How much is it?

I don't think it takes long if we are talking people who are indisputably Vietnamese citizens and indisputably relatives of the one whose name is in the book as head of the household.  If the mother has no close relatives in Vietnam and is no longer a VN citizen, it could be difficult.

Check this thread on a similar subject:  https://www.expat.com/forum/viewtopic.php?id=527741
In particular check post #6 by Budman1  The agency at the link he provides http://kieubaoviet.vn/   can help the mother and the baby but probably won't be much help for you.