It depends on what you specifically mean by "deconstructing" as I've heard it used to mean different things.
If you mean it in the most literal sense, which is picking at the character's personality to understand them better, there's a lot of different things you can do.
For me, giving them a character arc is the first step. What is their journey? How are they going to be different at the end of the story, to how they were before? If it's hard to answer those questions then you'll need to roll those questions around in your head. After all, if they have no arc then why are they performing the actions they do in the story, instead of another character? They have to have some kind of stake in the plot somehow to fit into the cast.
After I've decided on their literal character arc I like to try and translate that into a parallel emotional or psychological arc. Through the changes they undergo, what do they learn? How does their personality change? Is their worldview challenged somehow? Do they gain a new understanding of their self?
Then comes vicious questioning. I like to be asked loads of questions about my characters. The more specific the better. Me and my co-writer do this a lot. We will bring up one of our characters and then proceed to let the other ask as many questions as they can, almost like an interrogation. The idea is to condition yourself to pull out these details and nuances about your character, by thinking on the spot. It's a good idea to make notes of any revelations you have.
I actually created a free question course for you to do on your own if you don't have someone to workshop with - it's a work in progress but there are lists of detailed questions you can put your character through to train yourself in deconstructing and picking at their personality. It's really experimental and so far many have said it works. I haven't gotten any negative feedback on it yet, but I don't know whether or not it works for everyone. All I know is that for me and the few who have given feedback it definitely does.
If these springboards help, that's great. Maybe you'll find a better way of doing it, though.
