Question: > 
> How do you like your coffee?
> 
> 
> 

Can I use the above sentence either on an offering situation that I ask if they like it black or with sugar, or during having situation that I ask if the coffee tastes good?

When the latter is possible, can next both answers be possible? 

1. I like it a lot.
2. It's good.

Answer: 
You may use this in both situations, and indeed in others; the context will make it clear which meaning is intended:

* If you ask before you start preparing the coffee, you are asking how your hearer prefers their coffee to be prepared—Do you like it strong or weak (or possibly boiled to a black sludge)?
* If you ask before or as you serve the coffee, you are asking how your hearer 'takes' their coffee—Do you want cream or sugar with your coffee?
* If you ask after your hearer has started drinking the coffee, you are asking about the taste of the coffee—Is the coffee OK?

As for your answers, both are possible, as are many more: 

* It's terrific!
* It's horrible!
* It's coffee!
* It's Blue Mountain, apparently from St Thomas Parish; rather underroasted, to my mind, and a coarser grind should be used with that press.
This phrase does have two meanings in US English, as agreed by other posters.

For the sake of completeness, I feel compelled to add that in British English, this is a phrase that would only be used when offering coffee. Using it as a question to affirm how much someone was enjoying their coffee would probably be understood, but would sound extremely odd. Instead, British people might say:

> 
> "Are you enjoying your coffee?"
> 
> 
> "Is that coffee OK for you?"
> 
> 
> 

Or something similar.

Divided by a common language and all that.