In-network caching is one of the key features of information-centric networks (ICN), where forwarding entities in a network are equipped with memory with which they can temporarily store contents and satisfy en route requests. Exploiting in-network caching, therefore, presents the challenge of efficiently coordinating the forwarding of requests with the volatile cache states at the routers. In this paper, we address information-centric networks and consider in-network caching specifically for Named Data Networking (NDN) architectures. Our proposal departs from the forwarding algorithms which primarily use links that have been selected by the routing protocol for probing and forwarding. We propose a novel adaptive forwarding strategy using reinforcement learning with the random neural network (NDNFS-RLRNN), which leverages the routing information and actively seeks new delivery paths in a controlled way. Our simulations show that NDNFS-RLRNN achieves better delivery performance than a strategy that uses fixed paths from the routing layer and a more efficient performance than a strategy that retrieves contents from the nearest caches by flooding requests.