Quaternions and Rotations

John Stillwell's book Naive Lie Theory motivates the subject in the first chapter using an exploration of quaternions acting by conjugation on the unit sphere. I think his explanation is great for someone at any point in their undergraduate career, however the subsequent chapters may be extremely slow for anyone already acquainted with lie theory, group theory or general topology.

http://www.amazon.com/Naive-Theory-Undergraduate-Texts-Mathematics/dp/0387782141

There is a much more in depth coverage of the details of this in another book which I am trying to recall the name of.

edit: the book Finite Möbius Groups, Minimal Immersions of Spheres, and Moduli by Toth begins with a chapter on platonic solids and finite rotation groups all very explicitly, in terms of Möbius transformations. Once you understand how to interchange between Möbius transforms and quaternions acting on $R^3$ from stillwell, this may be helpful.