Recently YouTube has changed their process around G+ verification. The process is now automated and when your channel meets the necessary requirementsyou will automatically see the check mark appear. If you don't see it this means that you still have not met the requirements and you will have to wait.
Requirements:
Be connected with a Google+ page or profile

  • Your Google+ page or profile must be authorized by your commercial entity, organization, brand, or product.
  • If the channel is connected to a Google+ page, the page must contain a link to your organization’s verified website.
  • Have a substantial number of subscribers.

If you (channel owner/manager) have done any changes to your channels (changing username, email address associated with the channel) after your channel is verified, you will lose the check-mark. So please be sure to not make any changes to your channel identity after your channel has been verified.
More info from Google: https://support.google.com/youtube/answer/3046484