You could look at surfaces with maximal Picard rank. A surface is said to have maximal Picard rank if $H^{1,1}(X) \cap H^1(X, \mathbb{R})$ is spanned by curve classes. So the Kahler cone is the same as the ample cone, and you can compute the intersection pairing on $H^{1,1}$ by just intersecting curves with each other.

Now, this raises the question of whether there is a surface of general type with maximal Picard rank. Some googling found me the following:

Section 3 of [Quintic surfaces with maximum and other Picard numbers][2] shows that the Fermat sextic $w^6+x^6+y^6+z^6=0$ has maximal Picard rank. Apparently, this was originally computed by Beauville.

[Roulleau shows][3] that there are infinitely many Fano surfaces with maximal Picard rank, and gives a basis for the Neron-Severi group. (Confusingly, the [Fano surface][4] is not a Fano variety, but rather is of general type.)

A warning: If you look at [this paper][1] or [this one][2], they compute a lot of surfaces with maximal Picard. However, their method is to make a surface with an $A_n$ singularity and prove that its resolution has maximal Picard. Since this resolution has a $-2$ curve, it is automatically not of general type.



  [1]: http://www.springerlink.com/content/g55236l64258418w/
  [2]: http://arxiv.org/abs/0812.3519
  [3]: http://arxiv.org/abs/1001.4855
  [4]: http://en.wikipedia.org/wiki/Fano_surface