Without an image of the dog tag in question I can't be sure which one you are referring to, but it sounds like the "4th floor dog tag":
You can see it on your Battlelog profile in the section for right dog tags under the "exclusive" category.
There is a lot of speculation on how to get the dog tag, but it appears that there is nothing you can do anymore to get it. An answer from user MorningTimeCloud on the EA forum suggests that you needed to play the Battlefield Hardline beta:
If you played the [Battlefield Hardline] beta, you get the dogtag.
If you didn't then you won't get it.
I also found a Google Doc about the 4th floor dog tag which contains a lot of research about obtaining the tag and agrees with the EA forum. Here are the most relevant quotes:
just got off the phone to EA support
anyone else can freely do this to confirm
the 4th floor dogtag is NOT premium related and is NOT phantom program related, however it is HARDLINE related, they could not specify how its hardline related as it's something you need to do, and they will not reveal spoilers.
This isn't really new, but I asked on the Battlefield FB page and they
responded with this:
"The 4th Floor Dog Tag awarded to Battlefield 4 players is from the
Battlefield Hardline Beta. Those who successfully signed up for the
Battlefield Hardline beta through the beta sign-up site were awarded
the tag. If you received a code from a different source and did not
sign up through the site you will not receive the dog tag."