Read up on the Harry Potter universe, starting with the seven core books and branching out into auxiliary texts like "The Harry Potter Schoolbooks." Fan sites such as www.the-leaky-cauldron.org are invaluable sources of information, too. Instead of focusing on the plot, examine the details of the world itself, and what kind of character interests you.
Decide what type of person or creature your character should be. The default character in the Potterverse is a witch or wizard, but many other types exist. You may want to create a Muggle---a non-magical person, unaware of the wizarding world until happenstance reveals it to him. You could also create a centaur---wise astronomers in the forest around Hogwarts---or a pesky house elf working for a wizard family as a servant. You may even want to create a scary monster such as a Dementor. The selection will determine what kind of role he plays in the Potterverse and the sorts of experiences he might have.
Determine where in the Potterverse your character lives and works. For most people, that means being a student or a teacher at Hogwarts---the magical school where Harry and his friends learn their craft. But your character might live and work elsewhere---in the Auror's Office hunting down Death Eaters, for example, or in Gringotts bank handling goblin gold. Someone interested in the dark side of the world could create a character dwelling in Azkaban, either as a prisoner or some kind of administrator.
Write a background for your new character: her past, her upbringing, her skills and abilities, and her personality. It provides the character with a sense of weight and depth. Make sure you include things particular to the Potterverse; for example, if your character goes to Hogwarts, determine what house she's in and which professors are her favorites.
Draw a picture of your character if you're so inclined. It helps cement his image in your head. If you don't have a talent for drawing, use other methods for determining his looks: such as deciding which actor should play him in a movie version of his adventures.