The most obvious way to become one of the paprazzi is to simple follow celebrities around in their daily lives and submit embarrassing or revealing photos to various tabloids or newspapers. This could result in getting hired by those publications as well.
Start a "lifestyle" blog and start writing about public figures. There are several free sites you can also set up. Use a catchy title. Publicize it via social networking sites.
Of course, you can always see if you can get hired by one of the gossip magazines, but it's likely easier to just start your own.
Anyone can get a camera and hang out where celebrities are. The challenge is selling the photos in a timely fashion - make contacts to sell to before taking pictures.
You need to start taking pictures when you see famous people. Get a good portfolio put together and talk to the bosses at some well known magazines or newspapers.
You will need gear, good location and inside scoop before you begin your carreer as paparazzi. Research to find out where celebrities hang out and you have to be sneaky.