I like the book 'Geometrical methods of mathematical physics' by Bernard Schutz:

http://books.google.co.uk/books?id=HAPMB2e643kC

I'd say that the approach is fairly close to what the OP asked for.  I'm also a big fan of 'A first course in general relativity' by the same author:

http://books.google.co.uk/books?id=qhDFuWbLlgQC