The elements of the gradient of the local energy are then (using the chain rule and the hermiticity of the Hamiltonian) $$ \bar{E}_{i}= 2\left( \langle \frac{\bar{\Psi}_{i}}{\Psi}E_L\rangle -\langle \frac{\bar{\Psi}_{i}}{\Psi}\rangle\langle E_L \rangle\right). $$ From a computational point of view it means that you need to compute the expectation values of $$ \langle \frac{\bar{\Psi}_{i}}{\Psi}E_L\rangle, $$ and $$ \langle \frac{\bar{\Psi}_{i}}{\Psi}\rangle\langle E_L\rangle $$ These integrals are evaluted using MC intergration (with all its possible error sources). We can then use methods like stochastic gradient or other minimization methods to find the optimal variational parameters (I don't discuss this topic here, but these methods are very important in ML).