= Solution
Assume first that $f\ne0$ and let
$$
d=\operatorname{ord}_{(x,y)}f
$$
be the least total degree of a nonzero homogeneous part of $f$. The <associated graded ring> is
$$
\operatorname{gr}_{\mathfrak m}A
\simeq k[x,y]/(f_d),
$$
where $f_d$ is the initial homogeneous form. Multiplication by the nonzero polynomial $f_d$ is injective in $k[x,y]$, so the degree-$j$ component has dimension
$$
\dim_k(\operatorname{gr}_{\mathfrak m}A)_j
=\begin{cases}j+1,&j<d,\\d,&j\geq d.\end{cases}
$$
Summing the components of degrees below $n$ gives
$$
\boxed{
\chi(A,\mathfrak m;n)=
\begin{cases}
n(n+1)/2,&n\leq d,\\
dn-d(d-1)/2,&n\geq d.
\end{cases}}
$$
Thus the <Hilbert polynomial> is
$$
\boxed{P(n)=dn-\frac{d(d-1)}2.}
$$
Its leading coefficient is the order of vanishing, or multiplicity, of the plane curve $f=0$ at the origin. If $f=0$, no relation is imposed and $\chi=n(n+1)/2$ for every $n$.
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