The recently completed High Altitude Water Cherenkov (HAWC) gamma-ray observatory has been taking data with a partial array for more than one year and is now operating with >95% duty cycle in its full configuration. With an instantaneous field of view of 2 sr, two-thirds of the sky is surveyed every day at gamma-ray energies between approximately 100 GeV and 100 TeV. Any source location in the field of view can be monitored each day, with an exposure of up to $sim$ 6 hours. These unprecedented observational capabilities allow us to continuously scan the highly variable extra-galactic gamma-ray sky. By monitoring the flaring behavior of Active Galactic Nuclei we aim to significantly increase the observational data base for characterizing particle acceleration mechanisms in these sources and for studying cosmological properties like the extra-galactic background light. In this work we present first studies of data taken between June 2013 and July 2014 with a partial array configuration. Flux light curves, binned in week-long intervals, for the TeV-emitting blazars Markarian 421 and 501 are discussed with respect to indications of flaring states and we highlight coincident multi-wavelength observations. Results for both sources show indications of gamma-ray flare observations and demonstrate that a water Cherenkov detector can monitor TeV-scale variability of extra-galactic sources on weekly time scales. The analysis methods presented here can provide daily flux measurements with a minimum time interval of one transit and will be applied to new data from the completed HAWC array for monitoring of blazars and other transients.