How can I see my ping in Overwatch?

Enable the option "Display performance stats" in the video options.

This will show FPS and ping, in the top left corner.

Expanding on Kevin van der Velden's answer, the "Video" options were changed in an update to allow showing or hiding individual performance stats.

After enabling "Display Performance Stats", open the "Advanced Performance Stats" section below and make sure that "Show Network Latency" is enabled.

In addition to displaying ping, the section also has options for "Framerate", "GPU Temperature", "VRAM Usage", and "Network Interpolation Delay"