Can family use the same premium account?

I don't know how the premium features sharing would work- I presume you could figure it out-- but they don't allow players to have multiple accounts.  If you are all sharing passwords and logging in to the same account for videos or lessons, while presumably having individual accounts for games, you would appear as such an individual to the chess.com server. Sooner or later, one or more accounts would come to the attention of moderators, and one or more accounts would be banned.

Here is one sad account:

https://www.chess.com/forum/view/help-support/why-my-account-and-from-my-family-are-closed-and-my-ip-is-banned-for-no-reason2

I would write to support@chess.com to see if they have figured out a way for you to do this.

Can I share computers with another member?
https://support.chess.com/article/434-can-i-share-computers-with-another-member

You can share a computer, but each person must have their own account.