Integrated use of dual-energy computed tomography (DECT) with positron emission tomography (PET) has many potential clinical applications. However, the integration would either require costly hardware upgrade or increase radiation dose on PET/CT scanners due to the need for an additional Xray CT scan. The recently proposed PET-enabled DECT method enables DECT imaging on PET/CT without requiring the second X-ray CT scan. It combines the already-existing low-energy X-ray CT image with a 511 keV {\gamma}-ray CT (gCT) image reconstructed from time-of-flight PET emission data. A kernelized attenuation and activity (KAA) estimation method has been developed for reconstructing the gCT image from PET but the method has not fully exploited the potential of image prior knowledge. In this work, we propose a neural KAA method by using neural network representation as a deep coefficient prior to improve the existing KAA method. The resulting maximum-likelihood neural network-based reconstruction problem can be efficiently solved by utilizing the theory of optimization transfer. Each iteration of the algorithm consists of three modular steps: PET activity image update, gCT image update, and neural-network learning in the image domain. This algorithm is guaranteed to monotonically increase the data likelihood. The results from computer simulation and real phantom data have demonstrated that the proposed neural KAA method can significantly improve gCT image quality and consequent multi-material decomposition as compared to other methods.