Jot down a brief list of the types of powers your persona might have. Perhaps you are super strong, can block bullets or have the ability to fly. Maybe you shoot heat out of your eyes or can summon lightning bolts to smite your enemies. Whatever powers you decide upon, they should be thematically unified. Don't just assign them randomly but join them together in a way that makes sense. In some cases, you may not want to have any superpowers--just a piece of technology (like Iron Man) or your own physical skills (like Batman or the heroes from the movie "Kick-Ass").
Decide how you got your powers. If you're a student, perhaps you participated in a lab experiment at school. If you do a lot of driving, maybe you got struck by an industrial waste drum that fell off a truck on the freeway. The accident can be as outlandish as you like, but it should tie in to the powers you wish to have; if you want to have telepathy, then don't make an origin that would create super strength.
Come up with a name for yourself, something which rolls off the tongue and sounds really cool. It should evoke your powers in some tangible way--don't call yourself the Weather Master if you don't control the weather--and make good copy for the media.
Design a costume to wear. If you're really dedicated, you can sew your own or hire a tailor to create one for you, but in most cases, you can simply draw a picture of it. Certain websites such as HeroMachine let you create images of superhero costumes almost instantly. Make sure you include a mask to hide your identity if you don't want the world knowing who you are.
Decide how your superhero persona will impact your life. Superheroes must patrol the streets to fight crime, which can take up a lot of time. You need to determine which hours in the day you spend as a superhero, how you go about locating criminals (police scanners are recommended), whether you'll be able to balance it with work and family life, and what steps you take to keep your identity a secret. It's a tricky balance: many superheroes like Spider-Man constantly struggle to save the world and still keep up with the demands of real life.