The term $\nabla_b V_c$ is a (0,2) tensor writing in the abstract index notation, when writing in full basis form it reads
\begin{equation}
\nabla_b V_c \;dx^b\otimes dx^c\;,
\end{equation}
Now the status of $\nabla_b V_c $ is a components it is a scalar function while $dx^b\otimes dx^c$ is a basis of (0,2)-tensor.
Then the double covariant derivative reads
\begin{equation}
\nabla \Big( \nabla_b V_c \;dx^b\otimes dx^c \Big)\;,
\end{equation}
where $$\nabla_b V_c \equiv \partial_b V_c -\Gamma_b{}^q{}_c V_q\;.$$
The Leibniz rule is needed in this step
\begin{eqnarray}
\nabla \Big( \nabla_b V_c \;dx^b\otimes dx^c \Big)&=& \nabla \Big( \nabla_b V_c \Big)\;dx^b\otimes dx^c + \nabla_b V_c \;\nabla \Big( dx^b \Big)\otimes dx^c + \nabla_b V_c \;dx^b\otimes \nabla \Big( dx^c \Big)\;,\\
&=& \nabla_m \Big( \overbrace{ \nabla_b V_c}^{a\; scalar} \Big)\;dx^m\otimes dx^b\otimes dx^c + \nabla_b V_c \;\times \Big(-\Gamma^b{}_n dx^n \Big)\otimes dx^c \\&&+ \nabla_b V_c \;dx^b\otimes \times \Big( -\Gamma^c{}_p dx^p \Big)\;,\\
&=& \nabla_m \Big( \overbrace{ \nabla_b V_c}^{a\; scalar} \Big)\;dx^m\otimes dx^b\otimes dx^c -\Gamma^b{}_n \nabla_b V_c \; dx^n \otimes dx^c \\&& -\Gamma^c{}_p \nabla_b V_c \;dx^b\otimes dx^p \;,\\
&=&\partial_m \Big( \overbrace{ \nabla_b V_c}^{a\; scalar} \Big)\;dx^m\otimes dx^b\otimes dx^c -\Gamma_r{}^b{}_n dx^{r} \otimes \nabla_b V_c \; dx^{n} \otimes dx^c \\&&-\Gamma_s{}^c{}_p dx^{s} \otimes \nabla_b V_c \;dx^b \otimes dx^{p} \;,\\
&=&\partial_m \Big( \nabla_b V_c \Big)\;dx^m\otimes dx^b\otimes dx^c - \Gamma_r{}^b{}_n\nabla_b V_c \; dx^r \otimes dx^n \otimes dx^c \\&&-\Gamma_s{}^c{}_p \nabla_b V_c \;dx^s\otimes dx^b\otimes dx^p \;,\\
&=&\Big[\partial_m \Big( \nabla_b V_c \Big)- \Gamma_m{}^d{}_b\nabla_d V_c-\Gamma_m{}^e{}_c \nabla_b V_e\Big ]\;dx^m\otimes dx^b\otimes dx^c \;.
\end{eqnarray}
Then we define
$$
\nabla \Big( \nabla_b V_c \;dx^b\otimes dx^c \Big)=:\nabla_m \nabla_b V_c \;dx^m \otimes dx^b\otimes dx^c
$$
Finally, in abstract index notation we have
$$
\nabla_m \nabla_b V_c \equiv \partial_m \Big( \nabla_b V_c \Big)- \Gamma_m{}^d{}_b\nabla_d V_c-\Gamma_m{}^e{}_c \nabla_b V_e
$$
For a vector field, $X^\mu{}_{;\nu} = \partial_\nu X^\mu + \Gamma^\mu{}_{\kappa\nu}X^\kappa.$
Because the covariant derivative should coincide with partial derivatives on scalars, and should satisfy the Leibniz rule, $$\nabla_\mu (X^\nu X_\nu) = (\nabla_\mu X^\nu) X_\nu + X^\nu (\nabla_\mu X_\nu) = \partial_\mu (X^\nu X_\nu) = (\partial_\mu X^\nu) X_\nu + X^\nu (\partial_\mu X_\nu)$$
we can conclude that for 1-forms, $$X_{\mu;\nu} = \partial_\nu X_\mu - \Gamma^{\kappa}{}_{\mu\nu}X_\kappa.$$
Now, because the covariant derivative should satisfy the Leibniz rule, for a simple tensor $X^\mu Y^\nu$, $$\nabla_\rho (X^\mu Y^\nu) = (\nabla_\rho X^\mu) Y^\nu + X^\mu (\nabla_\rho Y^\nu) = (\partial_\rho X^\mu) Y^\nu + X^\mu (\partial_\rho Y^\nu) +
\Gamma^{\mu}{}_{\kappa\rho}X^\kappa Y^\nu
+ \Gamma^{\nu}{}_{\kappa\rho}X^\mu \ Y^\kappa $$
$$ = \partial_\rho (X^\mu Y^\nu) + (\Gamma^{\mu}{}_{\kappa\rho}\delta^\nu_\lambda + \delta^\mu_\kappa\Gamma^\nu{}_{\lambda\rho} )X^\kappa Y^\lambda.
$$
Because any tensor is a linear combination of simple tensors, it holds for arbitrary (2,0)-tensors that $$\nabla_\rho T^{\mu\nu} = \partial_\rho T^{\mu\nu} + (\Gamma^{\mu}{}_{\kappa\rho}\delta^\nu_\lambda + \delta^\mu_\kappa\Gamma^\nu{}_{\lambda\rho} )T^{\kappa\lambda} = \partial_\rho T^{\mu\nu} + \Gamma^\mu{}_{\kappa\rho} T^{\kappa \nu} + \Gamma^\nu{}_{\kappa\rho}T^{\mu\kappa}. \tag 1$$
The generalization to tensors of arbitrary type is now clear, and can be summarized in the following rule:
For every up index, add a Christoffel symbol contracted with the tensor on that index. For every down index, subtract such a term instead.
Now, when taking second covariant derivatives, it has to be remembered that the Christoffel symbols are not constants, so one has to take derivatives of them, also. And, the first covarian derivative adds an index, so for the second step, we need to use the formulas for tensors of the appropriate type.
E.g., for a vector, $X^{\mu}{}_{;\nu}$ is a $(1,1)$-tensor, so
$$X^\mu{}_{;\nu\rho} = \nabla_\rho(\partial_\nu X^\mu + \Gamma^\mu{}_{\kappa\nu}X^\kappa).$$
Now, in the operand, neither term is tensorial on its own -- remember: partial derivatives and the Christoffel symbols are not tensors Thus, $\nabla_\rho \partial_\rho X^\mu$ doesn't strictly make sense -- the covariant derivative can only act on tensors. But we can proceed formally using the appropriate version of (1) for each term, and adding them up, we do get the a tensor. Anyway, that gives something like
$$X^\mu{}_{;\nu\rho} =
\partial_\rho \partial_\nu X^\mu +
\Gamma^\mu{}_{\kappa\nu,\rho}X^\kappa +
\Gamma^\mu{}_{\kappa\nu}\partial_\rho X^\kappa +
\Gamma^\mu{}_{\kappa\rho}(\partial_\nu X^\kappa +
\Gamma^\kappa{}_{\lambda\nu}X^\lambda)
- \Gamma^{\kappa}{}_{\nu\rho}(\partial^\kappa X^\mu
+ \Gamma^{\mu\lambda}X^\lambda). $$
It is simple but extremely tedious and involves an awful profusion of indices to extend the calculation to the second covariant derivative of a $(2,0)$-tensor.
Best Answer
The formalism is explained very well in Landau-Lifshitz, Vol. II, par. 92 (properties of the curvature tensor). The Riemann curvature tensor can be called the covariant exterior derivative of the connection. The exterior derivative is a generalisation of the gradient and curl operators.
You might also consider looking at the geometry in differential forms language. The connection is seen as a 1-form (to be integrated along a line, the corresponding index is supressed), resulting in a (2-index) transformation matrix.* The Riemann curvature tensor is seen as a 2-form (to be integrated over a surface), again with values in a (2-index) transformation matrix. By doing so, you see Stokes' theorem appear, since integrating the connection (1-form) along a closed lines yields the same result as integrating the Riemann curvature (2-form) over the enclosed surface. That's why the Riemann curvature (2-form) needs to be the covariant exterior derivative of the connection (1-form).
Literature: Nakahara, Geometry, Topology and physics, chap. 5.4 and 7.
*Precisely: a Lie-algebra valued 1-form.