I suggest you consult chapter 19 of the book "Numerical Recipes in fortran" (or whatever programming language you prefer) by Press, Teukolsky, Vetterling and Flannery.

There's really too much for me to discuss and you've given insufficient information for me to respond in this post.

What are the boundary conditions?

If periodic then it would probably be best to transform to Fourier space and then carry out the discretisation.

What is the form of K(x,y)? If you have an actual expression (rather than a numerical matrix representation) of it then it may be possible to transform the problem to something more tractable.

I believe it's most likely you'll need to transform to fourier space then use an iterative method such as Jacobi, Gauss-Seidel or SOR method.

Anyway, checkout Ch19 of the book and have a look on the internet for the iterative methods I just mentioned.