I am about to launch a fan site for my favorite actress. I'd like to include a couple of short video clips showing her performing go-go at a nightclub, plus some photos. Will I run into any copyright problems? Does it make a difference whether the site is for paying members only?
Generally, there is nothing wrong with making a website about a public figure. While regular people enjoy a right to privacy, that right has lesser weight for individuals who intentionally step into the limelight, such as politicians and performers. Therefore, you are unlikely to face liability for a "fan page" about a celebrity, so long as you do not take any unreasonable steps to invade the celeb's private life (for instance, following her around, posting photos of her children, publishing personal information, and so on).
As for the video and photos, as long as you recorded that media yourself, then you automatically own the copyright and have certain exclusive rights. No formal registration is required with the U.S. Copyright Office in order to have a copyright.
However, keep in mind that the venue at which you recorded the actress might have prohibited recordings. While some large concert venues do not care, and indeed encourage fans to post to social media, some other theaters or auditoriums may not allow them. You could anger the venue, invite legal action, or perhaps be banned.
The copyright rules are different if you yourself did not make the video or photos in question. If they came from another source, and you do not have permission from the creator, posting that media is likely to be classified as copyright infringement. If the copyright owners become aware of your violation, you might have to pay money damages and remove the infringing material. The fact that your site is for paying members only will not protect you from charges of infringement. To the contrary, the commercial use of their protected material would weigh against your ability to post it.
On top of this, the use of someone's image for purposes of financial gain can violate that person's "right of publicity," a doctrine that prohibits you from implying that person endorses your products or services.
In short, while there is no problem with creating a fan website, you must be mindful of violating copyrights, as well as the possibility of violating the rules of various performance venues.