From the BBC and likely nowhere else. Copyrights are automatic upon creation of an original work of authorship in tangible form. No registration is required. Indeed, in the UK where the BBC is located, there is not copyright registration system so you can't check it. That leaves you needing to check with the BBC if you want to check. However, you don't need to waste your time checking since copyright is automatic - you can assume everything about it is copyrighted. You also need to check on trademarks, since the BBC has a number of trademarks related to Dr. Who. http://tsdr.uspto.gov/#caseNumber=73651559&caseType=SERIAL_NO&searchType=statusSearch
I am not your lawyer and you are not my client. Free advice here is without recourse and any reliance thereupon is at your sole risk. This is done without compensation as a free public service. I am licensed in IL, MO, TX and I am a Reg. Pat. Atty. so advice in any other jurisdiction is strictly general advice and should be confirmed with an attorney licensed in that jurisdiction.
The safest thing you can do is to assume that the content has been properly copyrighted in the US - Dr. Who is extremely popular and valuable series and the BBC is one of the largest and well-funded content providers. I'd try to get permission to use the content and/or, if your use may be protected under the fair use doctrine, you may want to get an IP attorney to write you an opinion on that. Another possibility is that the content you want to use is no longer under copyright protection if it was aired long time ago and registration formalities were not strictly followed, but that's a long shot.
You should assume that any TV series is protected by copyright law. You should know that under the Berne convention, U.S. courts honor copyright laws of other member states, such as the UK. UK copyright protection is somewhat stronger that protection in the U.S. Thus, you cannot use any of the characters or copy, broadcast, stream or otherwise distribute any episodes of Dr. Who without a license to do so. To obtain a license and identify copyright owners, you would need to work through the BBC.
You can start by conducting a simple search at the website for "U.S. Copyright Office - Search Copyright Records" which is located at http://www.copyright.gov/records/
The problem here is that a difference exists between copyrights that are "registered," versus those that exist and are valid. It is not necessary for the copyright holder to actually register their copyright in the US, and as such, lack of a result is meaningless, though a listed registration would be extremely helpful to you.
Next, you could attempt to contact BBC directly.