The elements of the gradient of the local energy are
$$ \bar{E}_{i}= 2\left( \langle \frac{\bar{\Psi}_{i}}{\Psi}E_L\rangle -\langle \frac{\bar{\Psi}_{i}}{\Psi}\rangle\langle E_L \rangle\right). $$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). Use methods like stochastic gradient or other minimization methods to find the optimal parameters.