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I am reading P. Schneider's paper, Introduction to the Beilinson conjectures. Section 4 in this paper is something very formal about Chern classes. Personally I find some terminologies in the paper a little old, and I have some difficulties in following it. So I am wondering are there any elegantly and thoroughly written references for the treatment of Chern classes that covers section 4 in this paper?

I guess elegant and thorough depends on my backgrounds. I know some algebraic K theory and motivic cohomology, but not much. Thank you!

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A reference I know for this is an article of Pushin: http://www.math.uiuc.edu/K-theory/0536/ and uses some mild motivic homotopy theory.

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