You already have Jesse's corrections, but I would like to focus on some of the wording he overlooked which, while grammatically correct, is not quite natural or idiomatic.

1.  I hope I can draw a happy and positive family portrait to you but I just want to keep it real.

In Chinese, 希望 and 愿 are defined, in part, as "to wish, to hope". But in English, the two words are not always interchangeable.

To say you "hope" you can do something means that you are going to try to do something which might be possible, but are not sure if you will be successful.

To say you "wish" you could do something, means that it would be nice if it was possible to do it, but you know the truth is that it can't be done. 

Also here, the phrase "keep it real" is a very casual slang expression that doesn't fit in with the serious tone and subject matter of the piece. You should use either "realistic" or "reality".

Therefore, I might suggest this revision:

I wish I could draw a happy and positive family portrait for you, but I want to be realistic.

(Meaning: "It would so nice if I could give you a happy portrait of my family, but the reality is quite different.")

It tells the reader from the outset not to expect a "happy and positive" story.

2. My mother [cries] almost every time when she looks back to her previous married life.

This fine as it is, but "almost every time" is the definition of the word "often". Also, "look back to" can be used in some contexts, but when talking about past lives, we say "look back on".  Finally, the word "previous" is most often used to distinguish a current thing from a different thing that came before: a previous owner, a previous husband, a previous job. Your mom only has one married life, so you can speak of the various times of that life with words like "early", "earlier", "middle", "current", or "later", for example.

So may I suggest this:

My mother often cries when she looks back on her early married life.

3.  ... take the responsibility of our family ...

We say "take responsibility for" something (omit "the").

Your father did not take responsibility for the care of your family.

4. ... my mother still lives in past life's shadow.

I'm going to disagree slightly here with Jesse, and suggest a more natural wording would be:

... my mother still lives in the shadow of her past life.

To "live in the shadow" of something is a common idiom, and it's best to retain that syntax.

5. My father's temper is not good enough to take her attitude in so ...

The phrase "take her attitude in" is not how we would express this idea. You might say "to deal with" or "to accept". Also, it is your father who can't accept her attitude, not his temper, so we should rearrange the wording:

  My ill-tempered father cannot deal with her attitude, so ...

To become familiar with natural sounding syntax and idomatic expressions is the next step in the improvement of your English skills. It's a big task, but fortunately, you are still a very young woman :)

Family Portrait- my parents

My father is 64 years old and mother is 58 years old. I hope I can draw a happy and positive family portrait to you but I just want to keep it real. I think mother loves father but they don't get along so very well together. My mother cryies almost every time when she looks back to her previous married life. She keeps accusing that my father didn't take the responsibility of our family so she had to do it for him, my mother left our hometown to find a job to support her three children and while my father stayed home to take care of us. This is was the start of conflict between them.

Life is much better now but my mother still lives in past life's shadow her past life's shadow. She is fragile and aggressive. My father's temper is not good enough to take her attitude in so they still quarrel with each other oftentimes.

That sounds very sad, all you can do to help is to give them both love and support.

By the way, your English is good but sometimes you mix-up your tenses. :)