Action Details
Requirements and how-to articles for YouTube's action:

Once your channel has 100,000 subscribers, you can submit a request to YouTube for channel verification. (This is different from verifying your account.)

You need this to get it done:

Your channel must have at least 100,000 subscribers in order to be able to apply for channel verification.
After submitting your application, YouTube will review your channel. YouTube verifies channels that:
Are authentic: Your channel must represent the real creator, brand or entity that it claims to be. YouTube will check different factors to help verify your identity, and may also ask for more information or documentation.
Are complete: Your channel must be public and have a channel banner description, channel icon and content, and be active on YouTube.
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How-To Articles
A list of useful links for you:

YouTube Help - Verification badges on channels
Information about verification, and the requirements for submitting a verification request.
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