While you might be able to make and actually obtain a copyright in a screenplay and/or completed film about a living or dead celebrity, you should be aware there are other rights that may seriously undermine your ability to distribute that film. Most important among these are the right of publicity (essentially the right of oneself to control the use of one's image).
A living celebrity will certainly be able to use this right (and nearly each state has its own version of right of publicity) against you in a fashion that blocks your distribution and/or extracts monetary damages from you. There are also potential legal issues such as defamation and trademark infringement that could arise from your making a movie without proper authorization from the celebrity. I would strongly advise you not to invest the time/money/energy in that process without securing agreements and releases that provide a solid footing to embark on the project. That being said, ultimately, you may be able to enforce the copyright against a third party that infringes your movie/screenplay (you just won't be able to likely profit from the movie yourself either).
I hope this helps.
Disclaimer: This answer is for informational purposes only and does not constitute general or specific legal advice, nor create an attorney client relationship.
CA has a very pro-celebrity publicity rights law, Civil Code section 3344, which also has a subsection 3344,1 for dead celebrities (i.e., their heirs). It gives only the individual the right to profit from their name, image, voice, photograph, signature, etc.
This law provides for disgorgement of profits, punitive damages, and attorneys' fees. it does have a few limitations for sports broadcasts, news accounts, and political campaigns, but not for films.
The right way to do a "biopic" is to get a "life rights" contract with the celebrity or the heirs.
But maybe there's already a life rights/publicity rights deal in place, if this movie already exists. If so, the celebritry knew they were being filmed and consented to it and signed a contract for their performance which included a grant of rights for use of their image for advertising for the movie, etc.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
As my colleagues have noted, the person about whom the movie was made enjoys a right of publicity that, at least at the outset, prevents his or her name and image from being commercially exploited w/o his or her permission.
But the movie is about a celebrity -- who is, by definition, in the public eye. The First Amendment to our federal Constitution trumps, in many situations, the celebrity's state-created right of publicity (a purely economic interest) because societal communications about matters of "public interest" are more important than the selfish interests of the celebrity.
There are, in addition, a number of California cases that either apply the common law and/or interpret the right of publicity statute which severely limit a celebrity's right of publicity and which expressly permit films to be made about the celebrity w/o the celebrity's consent.
Celebrities are famous for different reasons. Without knowing who the film is about and what the film is about no one can tell you whether you need the famous person's permission to make and/or distribute the film. You need to speak with an intellectual property attorney. Be sure to speak with one who is more than just a glorified talent agent and who can evaluate with some subtlety this issue and who has experience in "clearing the rights" to the content in films.
If the work is original and you own the copyright, the fact that others, their heirs or their licensees may have claims against you -- as the other responding attorneys here have explained, as usual, very comprehensively -- would not bar you from still enforcing your copyright in the work. But if you ultimately were required to disgorge your profits from that movie, or worse, such copyright enforcement may be a Pyhrric victory for you!