This paper proposes a set of tactile based skills to perform robotic cable routing operations for deformable linear objects (DLOs) characterized by considerable stiffness and constrained at both ends. In particular, tactile data are exploited to reconstruct the shape of the grasped portion of the DLO and to estimate the future local one. This information is exploited to obtain a grasping configuration aligned to the local shape of the DLO, starting from a rough initial grasping pose, and to follow the DLO's contour in the three-dimensional space. Taking into account the distance travelled along the arc length of the DLO, the robot can detect the cable segments that must be firmly grasped and inserted in intermediate clips, continuing then to slide along the contour until the next DLO's portion, that has to be clipped, is reached. The proposed skills are experimentally validated with an industrial robot on different DLOs in several configurations and on a cable routing use case.