In the last few years, we have seen an exponential increase in the number of Internet-enabled devices, which has resulted in popularity of fog and cloud computing among end users. End users expect high data rates coupled with secure data access for various applications executed either at the edge (fog computing) or in the core network (cloud computing). However, the bidirectional data flow between the end users and the devices located at either the edge or core may cause congestion at the cloud data centers, which are used mainly for data storage and data analytics. The high mobility of devices (e.g., vehicles) may also pose additional challenges with respect to data availability and processing at the core data centers. Hence, there is a need to have most of the resources available at the edge of the network to ensure the smooth execution of end-user applications. Considering the challenges of future user demands, we present an architecture that integrates cloud and fog computing in the 5G environment that works in collaboration with the advanced technologies such as SDN and NFV with the NSC model. The NSC service model helps to automate the virtual resources by chaining in a series for fast computing in both computing technologies. The proposed architecture also supports data analytics and management with respect to device mobility. Moreover, we also compare the core and edge computing with respect to the type of hypervisors, virtualization, security, and node heterogeneity. By focusing on nodes' heterogeneity at the edge or core in the 5G environment, we also present security challenges and possible types of attacks on the data shared between different devices in the 5G environment.