How to configure bind for a private subdomain?

Which dns server (resolver) do the computers in your local network use? If they use the internal bind server you administer, you can just set up the zone for your domain, and add the appropriate records for your internal services. The dns server will still work normally as a resolver for getting outside addresses, but will serve local zones to local users for your local domain.

Of course all the computers will have to use the dns server you set up for them, and not other dns servers (like opendns, google etc.)

You can configure Bind with split DNS. You would then have two separate zone files for the domain. The one to be served to the public would contain only the records you wish to have publicly available. The zone file for the same domain to be served internally would contain the same information plus any additional resources you want available to internal clients. This internal zone file could also serve the internal addresses for resources, rather than the public address, as appropriate.