In this paper, we introduced the concept of $C$-star bodies in a fixed pointed closed convex cone $C$ and studied the dual mixed volume for $C$-star bodies. For $C$-star bodies, we established the corresponding dual Brunn-Minkowski inequality, dual Minkowski inequality, and dual Aleksandrov-Fenchel inequality. Moveover, we found that the dual Brunn-Minkowski inequality for $C$-star bodies can strengthen the Brunn-Minkowski inequality for $C$-coconvex sets.