The various applications of the Internet of Things and the Internet of Vehicles impose high requirements on the network environment, such as bandwidth and delay. To meet low-latency requirements, the concept of mobile edge computing has been introduced. Through virtualisation technology, service providers can rent computation resources from the infrastructure of the network operator, whereas network operators can deploy all kinds of service functions (SFs) to the edge network to reduce network latency. However, how to appropriately deploy SFs to the edge of the network presents a problem. Apart from improving the network efficiency of edge computing service deployment, how to effectively reduce the cost of service deployment is also important to achieve a performance-cost balance. In this paper, we present a novel SF deployment management platform that allows users to dynamically deploy edge computing service applications with the lowest network latency and service deployment costs in edge computing network environments. We describe the platform design and system implementation in detail. The core platform component is an SF deployment simulator that allows users to compare various SF deployment strategies. We also design and implement a genetic algorithm-based service deployment algorithm for edge computing (GSDAE) in network environments. This method can reduce the average network latency for a client who accesses a certain service for multiple tenants that rent computing resources and subsequently reduce the associated SF deployment costs. We evaluate the proposed platform by conducting extensive experiments, and experiment results show that our platform has a practical use for the management and deployment of edge computing applications given its low latency and deployment costs not only in pure edge computing environments but also in mixed edge and cloud computing scenarios.