How to get the pdf of a joint distribution from its kernel? How could you obtain the pdf of a joint distribution from a multivariate kernel?

If you have the kernel of the joint density, all you are missing is the scaling constant that makes it integrate to one.  To obtain the latter, you need to integrate the kernel over the full range to get its definite integral.  The PDF is just the kernel divided through by the definite integral of the kernel --- i.e., if $k$  is the kernel and $\mathscr{X}$ is the range of the random vector then the PDF is:
$$f_\mathbf{X}(\mathbf{x}) = \frac{k(\mathbf{x})}{\int_\mathscr{X} k(\mathbf{x}) \ d \mathbf{x}}.$$
In many cases the integral in this equation will not have a closed-form expression and so you will need to approximate it with numerical methods or MCMC integration.
