I haven't thought about this in a long time. But it seems to me that the situation is like this:
The von Neumann algebra $\pi(A)''$ contains $A$ as a $\sigma$-weakly dense $*$-subalgebra. Then any normal linear functional on $M$ restricts to a bounded linear functional on $A$, so we have a map $\rho : \pi(A)''_* \to A^*$. This is a map of Banach spaces.
Then you use Kaplansky Density to show that $\rho$ is an isometry. And finally if you have a state on $A$, it is actually a vector state for $\pi(A)$, so it extends to a normal linear functional on $\pi(A)''$. Thus $\rho$ is an isomorphism of Banach spaces.
Then the dual map $\rho^* : A^{**} \to (\pi(A)''_*)^* \simeq \pi(A)''$ is also an isomorphism of Banach spaces. This allows us to define the algebraic operations in $A^{**}$ via transport of structure through the map $\rho^*$. You are asking how to define these operations intrinsically on $A^{**}$ without the use of the map $\rho^*$.
As you say, one way to understand this is to view $A$ as sitting weak-* densely in $A^{**}$ via Goldstine's Theorem. Then I guess you can try to extend the operations of $A$ to $A^{**}$ via continuity. Your question, then, is why does that definition agree with the operations on $A^{**}$ that we get from $\pi(A)''$?
I have not worked out the details, but it seems to me that it should work like this: we have an isomorphism of Banach spaces $\rho^* : A^{**} \to \pi(A)''$. But we know more, namely that this isomorphism comes from an isomorphism of preduals, so in fact this is an isomorphism of the weak-* topologies on these two spaces.
Both $A^{**}$ and $\pi(A)''$ contain $A$ as a weak-* dense subspace. The multiplication of $\pi(A)''$, when restricted to $A$, is exactly the original multiplication in $A$. And similarly for the multiplication of $A^{**}$ restricted to $A$ (since that's how we defined it). Thus we have a weak-* continuous isomorphism which is an isomorphism of algebras on a dense subspace, so it should be an isomorphism of von Neumann algebras.
There is another approach to defining an intrinsic multiplication on $A^{**}$, which is called Arens multiplication. I don't know much about it, but you can find the original paper "The adjoint of a bilinear operation" (Proceedings of the AMS 2, 1951) by Richard Arens freely available online.