Reading Serre's letter to Gray , I wonder if now modern expositions of the themes in Klein's book exist. Do you know any?

"Geometry of the Quintic" is available for free at my website. Jerry Shurman 


I got interested in this subject last year and just got round to writing up some notes which I hope may be of use. I also have a python script which implements the Klein's icosahedral solution of the quintic linked from this page. 


In Glimpses of algebra and geometry by Gabor Toth, chapter 25 is devoted to Klein's main result. 


Chapter 5 of McKean and Moll's "Elliptic Curves" explores the circle of ideas around Ikosaeder.I'm not sure if you'd consider this sufficiently "modern"  it's certainly a contemporary book but it doesn't use, say, schemetheoretic language. 


I covered Klein's "Lectures on the Icosahedron" in a modern way in my doctoral thesis: Elliptic Curves and Icosahedral Galois Representations, Stanford University (1999) http://www.math.purdue.edu/~egoins/notes/thesis.pdf A much shorter and more direct exposition is my publication in IMRN: Icosahedral $\mathbb Q$Curve Extensions, Mathematical Research Letters 10, 205–217 (2003) http://intlpress.com/site/pub/files/_fulltext/journals/mrl/2003/0010/0002/MRL20030010000200019947.pdf 


There is a (german) new edition of Klein's "Vorlesungen über das Ikosaeder ..." by Peter Slodowy (1993) with a large (about 80 pages) section of comments and remarks about new developments. 


There is the outrageously expensive "Geometry of the quintic" by Jerry Shurman, which discusses both Klein and DoyleMcMullen approaches (and then some more). 

