I was wondering if anyone had an explicit form of the hamilton-jacobi equations for GR. I had a little attempt myself using the 10 components of the metric as the 'co-ordinates', R*sqrt(det(g)) as the Langrangian density, but the maths got a bit messy when trying to compute their canonical conjugates.
The reason it was "messy" is because this approach obviously can't work. See Appendix E of Wald for a reasonably good treatment of the Hamiltonian formulation of GR. Once you've understood this material it should then be trivial to derive the Hamilton-Jacobi approach.