By a theorem of Matsusaka, every abelian variety $A$ over an algebraic closed field $k$ is a quotient of a Jacobian.
Now just apply Matsusaka's theorem to $A^{\vee}$, and dualize.
I think that the slightly weaker version where the field is infinite is proved somewhere in Milne's lecture notes on Abelian Varieties.
Reference.
T. Matsusaka: On a generating curve of an Abelian variety, Nat. Sci. Rep. Ochanomizu Univ. 3 1-4, (1952).
Quoting from P. Samuel review on MathSciNet:
An abelian variety $A$ is said to be generated by a variety $V$ (and a mapping $f$ of $V$ into $A$) if $A$ is the group generated by $f(V)$. It is proved that every abelian variety $A$ may be generated by a curve defined over the algebraic closure of $\mathrm{def}(A)$.