I found a few tools for video capture that are free and in one case work with Chromebooks and Google Drive.
Chromebook solution - Screencastify (Chrome Extension) and Convertio.co: First, install the Screencastify extension on chrome. Capture your video. Then save the video to GoogleDrive by clicking the Share button. This will save it as a “webm” file in a “Screencastify” folder in your google Drive so you need to convert it. Go to convertio.co and choose the open from GoogleDrive option AND the save converted files to Google Drive option. After you hit convert it will take a few mins to upload the file to Convertio, convert the filed, create a “Convertio” folder in your GoogleDrive and then upload the converted file from Convertio to that GoogleDrive folder.
Free windows software - TinyTake: This is a good free solution for making your screencast. It needs to be installed on your computer but will output mp4 files.
I can make a quick video walk through of the process if that is helpful