The increasingly complicated and diverse applications have distinct network performance demands, e.g., some desire high throughput while others require low latency. Traditional congestion controls (CC) have no perception of these demands. Consequently, literatures have explored the objective-specific algorithms, which are based on either offline training or online learning, to adapt to certain application demands. However, once generated, such algorithms are tailored to a specific performance objective function. Newly emerged performance demands in a changeable network environment require either expensive retraining (in the case of offline training), or manually redesigning a new objective function (in the case of online learning). To address this problem, we propose a novel architecture, DeepCC. It generates a CC agent that is generically applicable to a wide range of application requirements and network conditions. The key idea of DeepCC is to leverage both offline deep reinforcement learning and online fine-tuning. In the offline phase, instead of training towards a specific objective function, DeepCC trains its deep neural network model using multi-objective optimization. With the trained model, DeepCC offers near Pareto optimal policies w.r.t different user-specified trade-offs between throughput, delay, and loss rate without any redesigning or retraining. In addition, a quick online fine-tuning phase further helps DeepCC achieve the application-specific demands under dynamic network conditions. The simulation and real-world experiments show that DeepCC outperforms state-of-the-art schemes in a wide range of settings. DeepCC gains a higher target completion ratio of application requirements up to 67.4% than that of other schemes, even in an untrained environment.