Clique-width is a well-studied graph parameter owing to its use in understanding algorithmic traceability, and in this paper, we study the class of bounded clique-width graphs through the lens of succinct data structures. A data structure is said to be succinct if the amount of space used by the data structure is information-theoretically optimal up to lower-order additive terms. More specifically, we design a succinct data structure for graphs on n vertices whose clique-width is at most k≤ϵlogn for some constant 0<ϵ<1/6, that supports degree, adjacency, and neighborhood queries efficiently. This resolves an open problem of Kamali (Algorithmica-2018). As an application of our main technique, we also propose succinct data structures for distance-hereditary and Ptolemaic graphs, which are subclasses of the class of graphs with clique-width at most 3.