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.