Let q be the order of your finite field. Then the category of abelian varieties over $\mathbb{F}_q$ up to isogeny is semisimple - any object is isogenous to a product of simple ones in an essentially unique way, so this reduces your question to one about simple objects.
For simple abelian varieties over $\mathbb{F}_q$, there is the Tate-Honda classification which states that the isogeny classes are in bijective correspondence with Weil $q$-integers (algebraic numbers that have absolute value $q^{1/2}$ in all complex embeddings) up to Galois conjugacy.
I learned this from Milne's "Points on Shimura varieties mod $p$" (one of the articles from the Corvallis proceedings that Kevin mentioned), which has a nice and fairly elementary discussion in section 5. Hopefully somebody else knows the primary sources better.
