"this" pointer in C (not C++)

The typical approach in C is to have functions expect this as the first parameter.

int push(Stack *self, int val) 
{
  if (self->current_size == self->max_size -1) return 0;
  self->data[self->current_size++] = val;
  return 1;
}

This has the added benefit that, unless you need polymorphism, you don't need to put the functions in the stack, because you could just call push(stack, 10) instead of stack->push(stack,10).

C doesn't work like that. It's not an object oriented language. Functions that manipulate data structures need to take a pointer to the structure as an argument.

There's no implicit this in C. Make it explicit:

int push(Stack* self, int val) {
    if(self->current_size == self->max_size - 1)
            return 0;

    self->data[self->current_size] = val;
    (self->current_size)++;

    return 1;
}

You will of course have to pass the pointer to the struct into every call to push and similar methods.

This is essentially what the C++ compiler is doing for you when you define Stack as a class and push et al as methods.