The Feedback Vertex Set problem asks whether a graph contains $q$ vertices meeting all its cycles. This is not a local property, in the sense that we cannot check if $q$ vertices meet all cycles by looking only at their neighbors. Dynamic programming algorithms for problems based on non-local properties are usually more complicated. In this paper, given a graph $G$ of clique-width $cw$ and a $cw$-expression of $G$, we solve the Minimum Feedback Vertex Set problem in time $O(n^22^{O(cw \log cw)})$. Our algorithm applies dynamic programming on a so-called $k$-module decomposition of a graph, as defined by Rao \cite{R08}, which is easily derivable from a $k$-expression of the graph. The related notion of module-width of a graph is tightly linked to both clique-width and NLC-width, and in this paper we give an alternative equivalent characterization of module-width.