We introduce good ascending/descending filtration for a vertex algebra and show that the graded vector space of a vertex algebra associated to a good ascending/descending filtration is naturally a vertex Poisson algebra. As applications，we show that if the vertex Poisson algebra $gr_F(V)$ associated to a good ascending filtration F of V is non-degenerate, then V is non-degenerate. Furthermore, we show that if gr(V) is a free differential algebra, then V is non-degenerate. On the other hand, we construct a good descending filtration E of a vertex algebra and we show that the degree zero subalgebra of $gr_E(V)$ coincides with Zhu's Poisson algebra $V/C_2(V)$. Furthermore, we show that if $V/C_2(V)$ as an algebra is finitely generated, then V is generated by a finite subset with a PBW-type spanning property.