The question is in the title, so let me just repeat it:
How much information about $R-\mathrm{mod}$ can be extracted from $\underline{R-\mathrm{mod}}$ and $K_0(R)$?
Here $\underline{R-\mathrm{mod}}$ stands for the stable module category, i.e. $R-\mathrm{mod}$ after killing all projective modules (and necessary morphisms). Let's say $R-\mathrm{mod}$ denotes the category of finitely presented modules.
I realize this is an open-ended question. I started with a more concrete one: Whether one can reconstruct (up to, say, Morita equivalence) $R-\mathrm{mod}$ using the two data. Obviously, this is not possible, since both the data are trivial ($R-\mathrm{mod}$ is some "$0$-category" and $K_0(R)=\mathbb{Z}$) for, say, two different fields, which are not Morita equivalent (never, as Qiaochu Yuan points out in comments).
I should probably mention that this question is not really motivated by anything - just on the first glance, it seems that, if one thinks of $K_0(R)$ as "compressing the information about fin.-gen. projectives", together with the stable module category it should still contain a lot of information.
Thanks in advance for any help/suggestions.