You need only to establish zero field inside an infinitesimally thin homogeneous spherical shell. The same result follows immediately for shells of finite thickness, as these can be regarded as nests of infinitesimal shells! Note that the density can vary from shell to shell without spoiling the result.
I'd then divide the infinitesimal shell into 'hoops' following lines of latitude. You should be able to form an integral to give you the field strength at an arbitrary fixed point on the common axis of the hoops. It's clear from the symmetry of the hoop that components of field perpendicular to this axis will cancel.
Hope this will get you started. If you find it difficult, take comfort from the fact that Newton himself got stuck over this for some time…
I've just had a go at this myself, and got stuck trying to get the integral of the fields in terms of a single variable (though I'm sure it can be done). And (unlike Newton) I have at my disposal all the slick notation and methods of 'modern' algebra and calculus! But adding the potentials due to the hoops was much easier, partly because potential is a scalar. The total potential came out to be independent of the position of the 'fixed' point, meaning that the field strength is zero.