Recording a video with the webcam on your Chromebook can be accomplished through the use of a free Chrome app called CaptureCast. CaptureCast, produced by Cattura Video, allows you to record the screen on your Chromebook as well as input from your webcam.
To record a video with the webcam on your Chromebook open CaptureCast in your browser then allow it to access your webcam and microphone. You can specify how high of a resolution you would like to use to capture your video. You can also choose your audio quality. If you have an external microphone connected to your Chromebook, make sure that you have it enabled before you start recording.