I will describe how Hochschild-type methods and iterated integrals can be used to define natural extensions of the Chern and Chern-Simons forms to differential forms on the free loop space. I'llexplain how these can be used to construct a refinement ofdifferential K-theory, by using a variation of the construction of differential K-theory given by Simons and Sullivan. This is joint work with Thomas Tradler and Mahmoud Zeinalian.