The Bloch-Kato conjecture is actually more precise than that. As mentioned by Hunter Brooks, there is indeed an $\ell$-adic Abel-Jacobi map
$\phi $\phi : Ch^j(V)_0 \to H^1(G_K,H^{2j-1}{\mathrm{et}}(V\times{K}\overline{K},\mathbf{Q}_{\ell})(j))$H^1(G_K,H^{2j-1}_{\mathrm{et}}(V\times_{K}\overline{K},\mathbf{Q}_{\ell})(j))$$
The map $\phi$ is also called the cycle class map and is defined for any field $K$, say of characteristic $0$.
Now if $K$ is a number field, the conjecture of Bloch-Kato predicts that
(1) $\phi$ takes values in $H^{1}_f(G_K,\cdot)$.
(2) $\phi \otimes \mathbf{Q}_\ell$ is an isomorphism.
In fact (1) is a purely local question : this is really a question about the Abel-Jacobi map associated to a variety over $\mathbf{Q}_p$ (where $p$ can be equal to $\ell$ or not). The conjecture (2) together with the Beilinson-Bloch conjecture implies the statement you mention about the order of vanishing.
You can find a good survey on the $\ell$-adic Abel-Jacobi map and more details about what is known here :
J. Nekovar, p-adic Abel-Jacobi maps and p-adic heights, In: The Arithmetic and Geometry of Algebraic Cycles (Banff, Canada, 1998), 367 - 379, CRM Proc. and Lect. Notes 24, Amer. Math. Soc., Providence, RI, 2000.
See also the "Conjecture $\mathrm{Mot}_\ell$" in the following article :
M. Flach, The Equivariant Tamagawa Number Conjecture : A Survey.
