Currently, self-driving, emerging as a key automatic application, has brought a huge potential for the provision of in-vehicle services (e.g., automatic path planning) to mitigate urban traffic congestion and enhance travel safety. To provide high-quality vehicular services with stringent delay constraints, edge computing (EC) enables resource-hungry self-driving vehicles (SDVs) to offload computation-intensive tasks to the edge servers (ESs). In addition, caching highly reusable contents decreases the redundant transmission time and improves the quality of services (QoS) of SDVs, which is envisioned as a supplement to the computation offloading. However, the high mobility and time-varying requests of SDVs make it challenging to provide reliable offloading decisions while guaranteeing the resource utilization of content caching. To this end, in this paper we propose a \underline{co}llaborative com\underline{p}utation offlo\underline{a}ding and \underline{c}ont\underline{e}nt caching method, named CoPace, by leveraging deep reinforcement learning (DRL) in EC for self-driving system. Specifically, we resort to a deep learning model to predict the future time-varying content popularity, taking into account the temporal-spatial attributes of requests. Moreover, a DRL-based algorithm is developed to jointly optimize the offloading and caching decisions, as well as the resource allocation (i.e., computing and communication resources) strategies. Extensive experiments with real-world datasets in Shanghai, China, are conducted to evaluate the performance, which demonstrates that CoPace is both effective and well-performed.