In fact there are several Hopf algebra structures on this algebra, mainly because it's one of the many occurence of the free commutative graded algebra with exactly one generator in each degree $A=\mathbb C[h_1,\dots,h_n,\dots]$, where $h_i$ is of degree $i$. Here the $h_i$'s can be either the elementary or the complete symmetric polynomials. Of course that's the grading which makes things interesting, otherwise it would just be the usual polynomial algebra on infinitely many variables.
Indeed, let $G$ be the group of formal power series with constant terme equal to 1. Then the algebra $O(G)$ of polynomial function on $G$ is generated by the linear maps $\lambda_k$ defined by $$\langle \lambda_k,1+\sum a_nX^n \rangle=a_k/k!$$
Letting $\lambda_k$ having degree $k$, then the map $h_k \rightarrow \lambda_k$ is an isomorphism of graded algebra. But being an algebra of function on a group, $O(G)$ has a natural Hopf algebra structure given by
$$\Delta(f)(a \otimes b)=f(ab)$$ and $$S(f)(a)=f(a^{-1})$$
If you think as the $h_i$'s as being the elementary symmetric polynomials, then this coproduct is the same as the coproduct of Dan's answer (if I'm not wrong). This is not just an abstract isomorphism, however, but if I remenber well it is reminiscent of the fact that coefficients of a polynomial are elementary symmetric functions of its roots.
But if you take instead the group of formal power series with zero constant term, whose multiplication is given by the composition of formal power series, then you get again the same graded algebra but the above formula leads to a non-commutative coproduct (leading to the so-called Faa di Bruno Hopf algebra).