Mexico City, the largest city in Mexico, is also one of the largest cities in the world. It has over 9 million inhabitants and concentrates the vast majority of government and business centers. In this work we describe algorithms that use anonymized location data from mobile devices to construct Mexico City's contact and mobility networks aiming to help the analysis of the city's complexity by understanding movement and physical interaction patterns between its inhabitants. We show the effectiveness and usefulness of our approach by building networks with data collected in February 2020 and performing a general descriptive analysis on them. We found that contact networks in Mexico City are very sparse, characterized by a largest connected component, and with a heavy-tailed degree distribution. On the other hand, we observed that paths conformed by the highest-degrree nodes of mobility networks resemble Mexico City's street network; moreover, we found interesting qualitative differences in the degree distribution of these networks between weekends and weekdays. We present these results along with the release of contact and mobility networks.