Presheaves as limits of representable functors?
You mean "colimit of representable presheaves", not limit. Any limits that C has are preserved by the Yoneda embedding. So if C is, say, a complete poset like • → •, so that it is small and has all limits, you won't be able to produce any non-representable presheaves by taking limits of representable ones.
The way to write any presheaf as a colimit of representables is, like all things Yoneda-related, somewhat tautological, and should be worked out for oneself; but anyways it's explained to some extent at this nlab page. Rather than write out formulas, I usually think of the example of simplicial sets: every simplicial set X can be formed as a colimit of its simplices, i.e., a diagram of representables which is indexed on the "category of simplices of X", whose objects are pairs (n, x) where n is in the indexing category and x is an object of Xn. The same works in any presheaf category.
This follows from the Yoneda lemma - probably my favourite way of thinking about this fact is via coends in the way described here by Todd Trimble, which I think makes it quite clear what is going on.
By the way, that desciption by Todd Trimble has meanwhile been prepared at nLab:co-Yoneda lemma (this is what the question above is about) and nLab:Day convolution (for the more general statement).