The ins and outs of Character Arcs could fill a book by themselves, but here are a few pointers.
Try to give a character just one Character Arc. There are ways to give a character more than one, but they're tricky. They'd also require more dialogue than most games have room for.
A character's growth should be rocky. It's okay if the character resists growing for instance, if the character is put in a situation where he or she should grow, but doesn't, instead clinging on to his or her old ways. Sometimes a character needs to be "hit over the head" a few times before they start growing.
In Star Wars Episode IV, for example, Han has lots of exposure to the ethical group of Luke, Obi-Wan, and Leia before he finally starts changing and becoming concerned for purposes other than his own personal financial problems.
Happy Events Cause Growth Too
Although it's always difficult and uncomfortable for the character to grow, some of the events that force a character to grow can be happy ones.
In Episode IV, for example, when Luke first learns to use his light saber and hit a little flying target while wearing his blastshield, it's a happy moment, but still one that pushes him along the path of his growth.
Growth in Increments
Usually, like Luke in Episode IV or Neo in The Matrix, a character is forced to face his or her FLBW many times throughout a story. In some of these situations, the character may not grow at all and may even get defensive. (Luke is this way in the beginning, when he protests to Obi-Wan that he's too insignificant to do anything about the Empire.) When growth does occur, it's often in increments. Usually, by the end of the story, the character reaches the end point of his or her Character Arc.
For example, let's say you have a female character who lacks ethics. She's caught doing something unethical. She doesn't need to grow at that moment. Instead of repenting, she might justify her actions.
Darkness Before Growth
Characters can go through some very dark periods (emotionally) before they emerge on the other end of their Character Arc.
For example, in the film Good Will Hunting, Will (played by Matt Damon) is a young man who is afraid to let anyone become close to him. This fear stems back to terrible physical abuse as a child by his foster father that made him, on an unconscious level, equate intimacy with pain. So terrified is he of closeness that he even pushes away the woman who loves him and his therapist who only wants to help him. Rather than grow, he retreats into a very bleak situation. By the end, however, he makes his way through his FLBW and learns to allow people to be close to him.
The game Max Payne is another example. In it, Max feels responsible for his wife's murder. This guilt propels him into bleaker and bleaker situations and states of mind. One level is even played inside one of his haunting nightmares twice. By the end, though, he does avenge his wife's death.
The designers and writer decided not to have him feel fully restored from his feeling of guilt, however, believing that a rosy ending of that nature would violate the noir feeling. There's some real bleakness to Max's circumstances at the end, and a certain amount of bleakness within Max himself.
To make the emotions at the end even more layered, this darkness is mixed in with a feeling of resolution from Max killing his wife's murderer. All these choices show a daring move by the designers and writer, and, I think, a sophistication by the players who embraced the game so widely.
A Mask to Hide Limitations
Some characters hide their FLBW behind a Mask (see Chapters 2.1, "NPC Interesting Techniques," and 2.2, "NPC Deepening Techniques"). If you've given a character a Mask, once they grow through their FLBW, they no longer need their Mask and it will disappear.
Let's reconsider our woman who lacks ethics. She puts up a Mask, or false front, that she's a respectable citizen. She goes out of her way to impress everyone of the civic contributions she's made. When she finally obtains ethics in the end, she can drop her constant promoting of what a wonderful civic contributor she is.
Imply Success, Don't State It
It's usually poor writing to have the character overtly state how he or she has grown at the end of the story. You wouldn't want Luke to say, "I didn't know who I was, but now I do." Instead, by the end, we should infer from a character's actions and dialogue that they've "made it" to the end of their Character Arc and overcome their FLBW.
Failed Character Arcs
Not every character we're "pulling for" necessarily needs to reach the end of their Character Arc. If they fail, they become a tragic character, doomed to live forever ruled by their fear, limitation, block, or wound. Characters who fail tragically in their Character Arc can bring a "down" feeling to the end of a story, so think carefully before doing this with one of the characters you want the player to like.
For example, the character begins as a coward and still chickens out in the big battle at the end, although we thought he was gradually becoming more courageous. That would be a tragic character.
The Consistency of Villains
Villains rarely have Character Arcs. If they change, it's usually for the worse.