The question is a bit vague on the meaning of "using the stack data structure" but I'm going to take an educated guess on a possible implementation. I'll also be assuming that we are interested in a min-heap, that the stack support the push, pop, and top operations, and that the min-heap should support the make-heap, push, pop, and top operations.
The data structure consists of single stack $S$ that will contain the elements of the heap.
Make-heap$(A)$: To build a heap from a set of elements $A$, sort $A$ in nonincreasing order and push the elements one by one into $S$ (according to their order). In this way the last element to be pushed into the stack will be one of the minimums of $A$. This requires $O(|A| \log |A|)$ time.
Top: Return the top element of $S$ (in $O(1)$ time).
Pop: Pop the top element $x$ from $S$ (in $O(1)$ time). Return $x$.
Push$(x)$: To push a new element $x$ into the min-heap proceed as follows: create a new temporary stack $T$; while the top element $y$ of $S$ is smaller than $x$, pop $y$ from $S$ and push it into $T$; Finally, push $x$ into $S$ and iteratively pop the elements from $T$ while pushing them into $S$. When $T$ is empty you are done. Sadly, this requires $O(|S|)$ time in the worst case.