  Let $A$ be an abelian surface over an algebraically closed field
$\overline{k}$ with an embedding $\overline{k}\hookrightarrow\mathbb{C}$. When
$A$ is isogenous to a product of elliptic curves, we describe a large
collection of pairwise non-isomorphic hyperelliptic curves mapping birationally
into $A$. For infinitely many integers $g\geq 2$, this collection has
infinitely many curves of genus $g$, and no two curves in the collection have
the same image under any isogeny from $A$. Using these hyperelliptic curves, we
find many rational equivalences in the Chow group of zero-cycles
$\text{CH}_0(A)$. We use these results to give some progress towards
Beilinson's conjecture for zero-cycles, which predicts that for a smooth
projective variety $X$ over $\overline{Q}$ the kernel of the Albanese map of
$X$ is zero.
