= Solution
Use <metric signature> $(-,+,+,+)$, $x^0=ct$, and retain terms linear in $\Phi/c^2$ or $\Psi/c^2$. The <inverse metric> is $g^{00}=-(1-2\Phi/c^2)$ and $g^{ij}=(1+2\Psi/c^2)\delta^{ij}$. In the <Christoffel symbol> formula
$$
\Gamma^\alpha{}_{\beta\gamma}=\frac12g^{\alpha\delta}(\partial_\beta g_{\delta\gamma}+\partial_\gamma g_{\delta\beta}-\partial_\delta g_{\beta\gamma}),
$$
the differentiated <metric perturbation> is already first order, so the prefactor can be replaced by the flat <inverse metric>. For example $\Gamma^0{}_{ij}=\tfrac12\partial_0g_{ij}=-\delta_{ij}\partial_0\Psi/c^2$ and $\Gamma^k{}_{00}=-\tfrac12\partial_kg_{00}=\partial_k\Phi/c^2$. Evaluating the remaining components gives
$$
\boxed{\begin{aligned}
\Gamma^0{}_{00}&=\frac{\partial_0\Phi}{c^2},&
\Gamma^0{}_{0i}&=\frac{\partial_i\Phi}{c^2},&
\Gamma^0{}_{ij}&=-\frac{\partial_0\Psi}{c^2}\delta_{ij},\\
\Gamma^k{}_{00}&=\frac{\partial_k\Phi}{c^2},&
\Gamma^k{}_{0i}&=-\frac{\partial_0\Psi}{c^2}\delta^k_i,&
\Gamma^k{}_{ij}&=-\frac{\partial_i\Psi\delta^k_j+\partial_j\Psi\delta^k_i-\partial^k\Psi\delta_{ij}}{c^2}.
\end{aligned}}
$$
The lower-index symmetry follows from the <torsion-free connection>. Products of these first-order <Christoffel symbols> are beyond the retained <weak-field approximation>.
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