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.