Getting into my iCloud account

My Apple tvOS version is 6.2.1

Presumably you model Apple TV is 2nd generation then.

If you have an older model Apple TV and also two-factor authentication, then you may be asked to sign in with an adjusted procedure: Type your password followed by the six-digit verification code directly into the password field. Additionally, there is a time limit, after which the code becomes invalid.

Try this: First enter the characters of your password (do not submit yet), then manually request a new verification code, then append that code to the password (no spaces or other separating characters), and then submit.

This may be 2 factor authentification issue. Please follow the steps below:

• Apple TV asks for Apple ID password, enter it.
• An authentication code will be sent to your iPhone or iPad or Mac.
• Apple TV asks for the Apple ID password again, enter it followed by the authentication code. For example if your Apple ID password is XYZ123 and the authentication code is 4567 enter XYZ1234567
• Do this quickly or it will time out!