general relativity - Step by step algorithm to solve Einstein's equations

I cannot completely understand what is a regular method to solve Einstein's equations in GR when there are no handy hints like spherical symmetry or time-independence.

E.g. how can one derive Schwarzschild metric starting from arbitrary coordinates $x^0, x^1, x^2, x^3$? I don't even understand the stress-energy tensor form in such a case - obviously it should be proportional to $\delta(x - x_0(s))$, where $x_0(s)$ is a parametrized particle's world-line, but if the metric is unknown in advance how do I get $x_0(s)$ without any a priori assumptions?