Lifelong sequential modeling (LSM) is becoming increasingly critical in social media recommendation systems for predicting the click-through rate (CTR) of items presented to users. Central to this process is the attention mechanism, which extracts interest representations with respect to candidate items from the user sequence. Typically, attention mechanisms operate in a point-wise manner, focusing solely on the relevance of individual items in the sequence to the candidate item. In contrast, context-aware LSM aims to also consider adjacent items in the user behavior sequence to better assess the importance of each item. In this paper, we propose the Context-Aware Interest Network (CAIN), which utilizes the Temporal Convolutional Network (TCN) to create context-aware representations for each item throughout the lifelong sequence. These enhanced representations are then used in the attention mechanism instead of the original item representations to derive context-aware interest representations. Building upon this TCN framework, we propose the Multi-Scope Interest Aggregator (MSIA) module, which incorporates multiple TCN layers and their corresponding attention modules to capture interest representations across varying context scopes. Furthermore, we introduce the Personalized Extractor Generation (PEG) module, which generates convolution filters based on users' basic profile features. These personalized filters are then used in the TCN layers instead of the original global filters to generate more user-specific representations. We conducted extensive experiments on both a public dataset and an industrial dataset from the WeChat Channels platform. The results demonstrate that CAIN outperforms existing methods in terms of prediction accuracy and online performance metrics.