I'm looking for a reference that gives an overview of the most important properties of Arakelov intersection theory (on arithmetic varieties of arbitrary dimension) and that describes basic properties of the Arakelov Chow ring. There's a similar MO question asking about survey articles on (classical) intersection theory, so I guess that I'm asking the same question, but for Arakelov intersection theory.
A good reference in my humble opinion is Bost's paper in Bourbaki:
Théorie de l'intersection et théorème de Riemann-Roch arithmétiques
Séminaire BOURBAKI. Novembre 1990. 43ème année, 1990-91, n° 731
Another reference would be Soule's book on Arakelov geometry "Lectures on Arakelov geometry" written with Abramovich, Burnol and Kramer.
Finally, I know you didn't ask this, but in the case of arithmetic surfaces there are more references (besides Faltings' "Calculus on arithmetic surfaces" and Arakelov's original paper). For example, Deligne's paper "le determinant de la cohomologie" and R. de Jong's Ph.D. thesis: http://www.math.leidenuniv.nl/~rdejong/publications/thesis.pdf Also, Moret-Bailly's paper "Metriques permises" in Szpiro's 1985 Asterisque is wonderful.