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Is there any literature that introduces hyperelliptic curves without the view towards cryptography? Even better is if there are any books that talk about them with (about) the same amount of detail as Silverman does for elliptic curves.

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For Genus $2$ there is a very nice book by J.W.S. Cassels and E.V. Flynn:

Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2

For higher genus I have only seen short sections in books on Riemann Surfaces and Abelian Varieties, mostly in the form of a more elaborate example for the general theory.

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I checked this out from the library today and it is a very clearly written book indeed. –  Eugene Jun 26 '12 at 2:52

There is a new book on Riemann surfaces which has the advantage that hyperelliptic curves are treated right at the beginning.
And the icing on the cake is that the publisher (Cambridge University Press) lets you read precisely those pages for free!
The authors frequently come back to hyperelliptic curves in order to illustrate the notions they introduce with non-trivial examples.
For example they give a very explicit description of the global differential forms on such a curve by exhibiting a basis of that complex vector space (pp.21-22).
And on pages 65-67 you will find highly non-trivial calculations (Hyperelliptic involution, Normalized model of hyperelliptic curves) which will familiarize you with these Riemann surfaces.

As an aside, let me point out that the book is quite up-to-date and ends with an introduction to Grothendieck's theory of dessins d'enfant.

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One of my friends recently recommended me a book that introduces hyperelliptic curves with a view towards number theory: Silverman and Hindry's Diophantine Geometry: An Introduction.

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