Adding an expiration date to a video will automatically set it to Unpublished, which will remove it from your catalog.
Videos scheduled to expire will be removed sometime between 12:00 AM UTC and 12:10 AM UTC on the date that you set it to be unpublished. For example, if you set the video to expire on April 14, then it will expire on April 14 between 12:00 AM UTC and 12:10 AM UTC.
To select the date:
Navigate to the video you want to set an expiration date for under Content > Videos and click Edit. Under the Publication tab, click either Published or Scheduled, add a date in the Expiration Date section, and then Save those changes.