The growing adoption of information and communication technologies (ICTs) is enabling intelligent power grid applications. However, strong reliance on ICTs makes the grid susceptible to cyber attacks such as false data injection attacks (FDIAs). This paper shows how deep learning approaches can be used to craft FDIAs against power grid state estimation that can circumvent the grid's bad data detector (BDD). In particular, we utilize conditional Generative Adversarial Networks (cGANs) to learn the distribution of the power grid measurement data and produce fake measurements that are identical in distribution to the real ones. Under the proposed algorithm, the attacker needs to have access to the grid's measurement data and know what data types in order to inject into the measurement system. No other prior knowledge about the grid is required. This type of threat model is novel and has not been considered so far. The simulation results on IEEE 14-bus system shows that FDIAs generated by our best performing cGAN implementation trained using real-world load data sets can bypass the BDD with a very high probability. Moreover, the distance between the distributions of the real and fake measurements (with FDIAs), measured in terms of the Jensen-Shannon divergence has a very low value, which shows the effectiveness of the proposed FDIA design approach.