In recent years diamond magnetometers based on the nitrogen-vacancy (NV) center have been of considerable interest for magnetometry applications at the nanoscale. An interesting application which is well suited for NV centers is the study of nanoscale magnetic phenomena in superconducting materials. We employ the magnetic sensitivity of NV centers in diamond to interrogate the magnetic properties of a thin-layer yttrium barium copper oxide (YBCO) superconductor. Using fluorescence-microscopy methods and samples integrated with an NV sensor on a microchip, we measure the temperature of phase transition in the layer to be 70.0(2) K, and the penetration field of vortices to be 46(4) G. We observe the pinning of the vortices in the layer at 65 K, and estimate their density after cooling the sample in a ~ 10 G field to be 0.45(1) μ m^{-2}. These measurements are done with a 10 nm thick NV layer, so that high spatial resolution may be enabled in the future. Based on these results, we anticipate that this magnetometer could be useful for imaging the structure and dynamics of vortices. As an outlook, we present a fabrication method for a superconductor chip designed for this purpose.