The first thing I do is put the character into a believable situation. The rest comes naturally.
For a protagonist to be believable they have to be hurt, to have feelings and to be able to master their feelings. For an antagonist to be believable they have to be resistant to pain, or numb. They must dehumanize themselves for an overambitious goal.
It helps if you use, as a model, someone you know. Unless it is biography, it is also good if you don't name the character after the person you know.
A bit of myself, a bit of others I know plus some of what I have imagined.
A connection with the reader. Even the most outrageous superheroes have qualities every human has. My characters are mostly based on people I've encountered in my own life. I draw great inspiration from my life experiences.
Lots of details, and frank statements about how characters feel about other characters. I create mine through many revisions. If I don't believe a character, I think of how to make the character believable, then go try and do that.
He has to be 'human' with all the strengths and weaknesses. I take them from real life.
I have found that if the character has something remotely in common with the readers then it makes your story more believable.
I look for the real life equivalent. I then do as much research as I possibly can on the real life equivalent of my fictional character. Basing the character on a real person always makes it easier to make the character stand out as believable.
It is unusual for me to make up a character out of thin air. They are individuals or composites of individuals with whom I have run across in my numerous travels in Asia, Africa,North America, South America, Europe, and Central Asia. I sometimes engaged in meaningful dialogue with many of them. I take copious notes.
Sometimes the plight of a country such as the Congo moves me to such an extent that the girls and young women I met and conversed with and danced and sang with, morphed into my head and then came out spontaneously, like Pallas Athena from her father Jove's head. That is how the heroine of my book Heart of Diamonds was born.
You must believe in your character and support her/him in every way. She must be larger than Life or even Death. And yet...she must have imperfections, a grisly history etc.
For a character to be believable I think it should have a solid backstory. It is up to me as a writer to flesh out the backstory in ways that let the readers know what constitutes the character. I like to focus on the minute details of the way a person interacts or if has a nickname, I like to know the origin of it.
A character has to breathe on the page and has to really want something, with great stakes attached. It takes me a long time to know my characters and I just have to be patient with the many drafts and the process.
I think a character should be multi faceted. For example, Lou, the protagonist of my first book Summerset grows into quite a hard faced bitch and yet we see her from childhood and all the knocks she takes to make her into the person she is. No one is perfect and no one is entirely bad so I like my characters to be three dimensional. I honestly cannot say how I create characters, they spawn from a dark place in my head and that's the Gods honest truth. Lou was probably the nearest to me (but I'm not a hard faced bitch I promise!).
For a character to be believeable they have to have flaws, to put it simply. Absolutely no one is perfect, I don't care how many writers love their Mary Sues/Stews, without a few flaws (a tragic one is even better) no one will buy into it. Characters are made with bits of ourselves, our hopes and fears, and if we're not willing to carve out a piece of our flesh and put it on the page for everyone to see, the story isn't worth reading.
If the character can create within the reader a sensation...a feeling, then he is believable. In this way I am an emotive writer...not psychological. If my characters are believable it is only because they are capable of creating a sensation and no other reason. My writing is sparse...the feelings I create, simple.