The first thing you need to think about when creating a character is determining which type of character it is you want to create. There are many different character types, from archetypes to stock types to more individualized characters. You'll first want to determine which type of character yours is.
Stock Types: A stock character is someone based on a common literary or social stereotypes. Stock characters rely heavily on cultural types or names for their personality, manner of speech, and other characteristics. In their most general form, stock characters are related to literary archetypes, but they are often more narrowly defined. Stock characters are a key component of genre (e.g. fiction, satire, romance, etc.), providing relationships and interactions that people familiar with the genre will recognize immediately. Stock characters make easy targets for parody, which will likely exaggerate any stereotypes associated with these characters.
Support Character: A supporting character is a character in a narrative that is not focused on by the primary storyline. Sometimes supporting characters may develop a complex back-story of their own, but this is usually in relation to the main character, rather than entirely independently. Supporting characters may appear in more than half of the episodes per season.
Protagonist: A protagonist is the main character of a literary, theatrical, cinematic, or musical narrative, who enters conflict because of the antagonist. The audience is intended to most identify with the protagonist.
Antagonist: An antagonist is a character, group of characters, or institution that represents the opposition against which the protagonist or protagonists must contend. In other words, an antagonist is a person or a group of people who oppose the main character(s).
In the classic style of stories wherein the action consists of a hero fighting a villain/enemy, the two can be regarded as protagonist and antagonist, respectively. Of course, some narratives cast the villain the protagonist role, with the opposing hero as the antagonist.
The antagonist may also represent a major threat or obstacle to the main character by their very existence, without necessarily deliberately targeting him or her.
For more information on character types and archetypes, go to tvtropes.org.
Now that you know what type of character you are building, you can begin to concept him/her. What kind of person are they? What is their personality? Their likes? Their dislikes? What are their strengths and weaknesses? Do they have a love interest? Or are they a lone wolf type? All of these traits define who they are as a person, which is important because you want the people reading your story to identify with each character in some way, whether it's simply identifying them as 'the bad guy' so they can have someone to hate, or the hero so they have someone to cheer for, or even reversing the roles and making them sympathetic to the bad guy and dislike the hero. The choice on how you want to build your character is yours.
When you have finally figured out who your character is, meaning how they act, what kind of person they are and everything else involved in concepting a character, you can begin fleshing out their physical appearance. Some people say that you should do this first, but I have always believed that a characters personality should determine what they look like. You couldn't picture someone like Naruto acting like Sasuke, could you? No? Didn't think so. This is why determining your characters physical appearance after you've build their personality is important.
Example, if you have a man whose personality is 'dark' and 'mysterious' then giving him bright blond hair and sapphire blue eyes probably won't give off a 'dark and mysterious' aura nearly as well as making him have raven colored hair and enigmatic green eyes. The way they act should determine what they look like. This will allow your readers to identify with the character even more.
So, now that you know how to build a character, let's talk about how this works in a fanfiction. Fanfiction is tricky because you are using characters that have already been created. Unless you're trying to make an OC character, then you have to work within the guidelines that the original creator of that character used to concept them.
One of the best ways to make a character your own in fanfiction is to force a gradual change through a combination of different plot devices that slowly and realistically change the characters personality.
A good example would be making your character watch the death of a loved one and slowly withdrawing from everyone else and pushing themselves to become stronger so they'll never have to witness a tragedy like that again.
Or, if you want a more comical route, force your character into situations that they are not used to in order to effect a change. A good example of this can be found in most Japanese Ecchi (perverted comedies), in which the Protagonist will often find him or herself in perverted situations that they would have never found themselves in before.
For more information on standard Japanese Ecchi scenes used for this purpose, you can go to tvtropes.org and look up tropes like the Shower of Awkward and Accidental Pervert.
If you have any specific questions about how to create a character, or if you want some extra help creating a character, post your question in this blog and I will get back to you.