This is a big question, and depending on the type of character arc you're writing you might have to alter your process, but here's a simplified look at the different facets of a character arc and how they work:
1. Figure out your starting and ending point
If either of these aren't clearly defined in the planning stages, your character arc isn't going to be comprehensive. Let's use an example for this; Macbeth, who has an interesting little arc which we can explore.
Macbeth begins the story as a war hero and an important and loyal member of King Duncan's court. He ends the story as a mad, tyrannical king responsible for dozens of murders. A pretty big shift; don't worry if it's too extreme at this point, as figuring out how they got from A to B is what we're doing next.
2. Establish the pre-existing factors which lead to the change
Nobody changes without reason, and a big life event isn't enough to make them change in a specific way. Different people will react to similar situations in different ways. For example, a person who has already experienced tragedy will react differently from someone who has never lost someone before.
These pre-existing factors may be internal or external, or a mix of both. Let's go back to our Macbeth example. The factors that lead Macbeth to kill Duncan are largely his ambition and previous accomplishments as thane and as a war hero. His lust for power as well as his pride at his past feats make him believe that being king will be beneficial for him.
Keep in mind that the nature of these factors is that they're not strong enough for the change to occur by them alone. Macbeth's ambition and accomplishments are still not enough to overpower his loyalty to Duncan and moral opposition to killing. That's where the next part comes in.
3. Create the inciting incident
What is the event that causes your character to pass the threshold and begin to change? What is the straw that breaks the camel's back, combining with the pre-existing factors to incite it?
For Macbeth, this is the witches' prophecy, and Lady M's subsequent urging to kill Duncan. Keep in mind that the inciting incident doesn't have to be a single static event; it can be an incident in which a variety of factors, including the pre-existing ones, push the character to begin to change.
Now's a good time to ask the question: What does your character want? This is the event that will convince them they can get it by acting a certain way.
4. Let your character act upon their new desires
So, they've changed, just a little, due to a couple factors. Now's the time for them to begin acting upon their change, to drive the plot and show the audience how their character is developing.
Wherever your character is headed, you can't make them immediately get there after the inciting incident. They need to progressively act upon their desires until their actions get more and more extreme. Generally, this is because by acting, they've begun to achieve what they desire, and this only adds onto their want as they've received a positive reaction. For Macbeth, this is the murders he commits.
For a positive character arc example, perhaps this is the character, after realising they need to be more open about their emotions, talking about their feelings with their friends and family. This causes a positive change in their life so they continue doing it.
5. End the arc
Now you've reached the ending point which you decided upon at the start. This is the 'new person' that they've become due to the change that has occurred and the actions they've taken. For Macbeth, he's now a tyrant king. His arc ends with him being killed by Macduff.
6. Ensure the arc is easily understood by the audience
Now that you've outlined it, you can begin to look over your arc. Make sure that you foreshadowing pre-existing factors before the inciting event so it doesn't feel like the change came out of nowhere. Dive into your character's psyche and make sure the audience understands their reasoning from their point of view, even if it's relatively sick and twisted.
That's a pretty basic way to write a character arc. This method won't work perfectly for all authors; experiment around with your character and their arc, and make sure to consider how the audience will interpret it. This is where big-picture thinking comes in, and most likely where the writers of Game of Thrones failed with Dany. By not making the arc easily understood by the audience, by making it hard for them to empathise with her and understand how she got to that place, the arc is badly written.
For further reading beyond this basic overview, there's a good series of articles by K.M. Weiland I'd recommend checking out: