Maybe the paper Lazar, Aldo J., and Joram Lindenstrauss. "Banach spaces whose duals are L 1 spaces and their representing matrices." Acta Mathematica 126.1 (1971): 165-193 will be helpful for you? Or this one: Lindenstrauss, Joram, and Daniel E. Wulbert. "On the classification of the Banach spaces whose duals are L1 spaces." Journal of Functional Analysis 4.3 (1969): 332-349.
I doubt it... the dual of C[0,1] is the space of all finite signed Borel measures on [0,1]. This is a non-separable space (no countable dense subset). If m is a bounded positive Borel measure on a compact set then L^1(m) is a separable space. So m would have to be rather weird for L^1(m) to be isometrically isomorphic to C[0,1]'.