The fifth generation (5G) of mobile networks is steering the technological evolution of many application domains, including many demanding automotive scenarios, such as platooning. Such evolution combines softwarization enablers (e.g., Network Function Virtualization (NFV) and Software-Defined Networking (SDN)) with Cellular Vehicle-to-Everything (C-V2X) communication to form end-to-end network slices with 5G infrastructure resources. Unlike the currently used 802.11p standard, the adoption of these technologies would support Ultra-Reliable Low-Latency Communication (URLLC) between the platoon members and would offer better network coverage, which in turn would allow faster remote monitoring and operation of the entire platoons. In this paper, we present a novel C-V2X network slicing framework for platooning applications. The proposed framework includes a library of C-V2X Virtual Network Functions (VNFs) that can be used to customize network slices spanning over the entire 5G infrastructure. The implementation also includes edge computing functions to provide further latency and performance improvements as well as network security monitoring functions to provide cyber-resilience for the platoon. We have conducted implementation tests on a real 5G infrastructure forming a truck platoon in an automotive campus. The results demonstrate the support of URLLC on the C-V2X connectivity between trucks, which opens new horizons towards the adoption of C-V2X network slices in future platooning applications.