How should I find the negation of this statement? I found that I need to find the negation of the following statement when I tried to give a proof of an exercise:

...For each given $x\in X$ and all $\epsilon>0$, there exists $n>0$ such that for all $y\in X$, there exists $k\in[0,n]$ such that $\|f_k(y)-x\|<\epsilon.$

Question 1
I'm confused by the phrase "such that" when I try to rewrite the statement by symbols:

$\forall x \in X\  \forall\epsilon>0\ \exists n>0\  \forall y\in X\ \exists k\in[0,n] (\|f_k(y)-x\|<\epsilon)$. 

I'm not sure if this is correct or not. 
Question 2
Besides, should the negation be

$\exists x \in X\ \exists\epsilon>0\forall n>0\exists y\in X\forall k\in[0,n](\|f_k(y)-x\|)\geq\epsilon)$?

Question 3
In general, how should one deal with a statement where there are many "such that"?
 A: "Such that" has no mathematical meaning, it simply expresses that the sentence is not over.
And you are right about your translation and the negation.
A: Think of 'there-exists-an-....-such-that' as a unit. 
It isn't quite right to say that 'such that' has no mathematical meaning here: better, it has no separate meaning, but should be thought of as part of the whole phrase which expresses the ordinary-language quantifier (and it's a part you typically can't omit while preserving grammaticality).
A: The way I remember this is that after an $\exists$ there's always a "such that" giving the property that the existing thing satisfies. So for your first question, since each "such that" comes right after an $\exists$, I'd just let them be implicitly given by the $\exists$. It looks to me like your interpretation is correct.
As to your second question, the way I think about it is "$\neg\forall x = \exists x\neg$" and "$\neg\exists x = \forall x\neg$." That is, "it's not true for everything" is the same as "there's something where it's not true" and "it's not true that it exists" means "for everything, it's not true."
So, symbolically, write $\neg$(expression), then move the $\neg$ through each quantifier until you get to the proposition you're quantifying, and then negate the proposition. Here, you basically have
$$\forall x\forall\epsilon \exists n\forall y \exists k \ \ P(k,n,x,y,\epsilon),$$
so its negation is
$$\neg\bigg( \forall x\forall\epsilon \forall n\exists y \forall k P(k,n,x,y,\epsilon)\bigg) = \exists x\exists\epsilon\forall n\exists y\forall k \ \ \neg P(k,n,x,y,\epsilon)$$
Alternately, and maybe more usefully for understanding the math, think about what the statement means ("for any $x$ and any precision level $\epsilon$, you can always go out far enough in the sequence so that for any $y$ you can find some function in the sequence before that point that takes $y$ $\epsilon$-close to $x$"), then negate it:

You can find some $x$ and some precision level $\epsilon$ such that, no matter how far out in the sequence you go, you can always find some $y$ that is taken more than $\epsilon$ away from $x$ by everything thus far in the sequence.

