If null set is an element of a set then will it belongs to set or subset? if $$A=\left\{\varnothing \right\}$$ we know that $A$ is a Singleton Set whose single element is a Null set. That is $A$  is a set containing another set which is Null set.
Now are these two statements True?
$1.$ $\varnothing \in A$
$2.$ $\varnothing \subset A$
 A: Both statements are true.  One can think of a set as a "bag".  Then $\emptyset$ means an empty bag.  In this case, $A=\{\emptyset\}$ means a bag containing only one element which is an empty bag.


*

*$\emptyset\in A$ since $A$ contains an empty bag inside it, the empty set is an element in $A$.

*$\emptyset\subset A$ since every element of the empty set (of which there are none) is also in $A$. 
A: The empty set is a subset of every set. It is also an element of your set.Thus  both statements are true.
A: As you say, "we know that $A$ is a Singleton Set whose single element is a Null set", so $\emptyset\in A$. Indeed, every set contains the empty set, so $\emptyset\subseteq A$ too. 
A: Elements
In the notation $A=\{\varnothing\}$ everything between the curly braces (except possible commas) is considered to be an element of the set, and we can denote this by
$$\varnothing\in A.$$
This is nothing special about the empty set. As I said, the curly braces enclose the elements, e.g. if $B=\{\color{red}\blacklozenge,\color{blue}\bullet, 7,\times\}$ then
$$\color{red}\blacklozenge\in B,\qquad \color{blue}\bullet\in B,\qquad 7\in B\qquad \text{and}\qquad \times\in B.$$

Subsets
The statement $\varnothing\subseteq A$ is always true no matter how the set looks like. This is because the empty set is a subset of all sets without exception. Subsets model the idea of "choosing" some of the elements, not necessarily all. And you have always the option to choose none, which gives $\varnothing$.
