I usually get my ideas by just taking showers. If I'm motivated enough to pursue what I have been thinking of, I type it down as soon as I possibly can and hope it sticks. But yeah, for me, it's either just "hey wouldn't it be cool if x did this thing" and I just try to roll with it from there, or me getting a sudden spur of motivation by consuming recent media, mainly other video games, and mimicking the writing style or the topics to the best of my abilities.
As for how I actually get into the character(s) I just made up... well, I usually try a light hearted trope - such as, "Cardamine tripping Lane Hacker that sometimes speaks in vague internet shitposts and humorous references", or "Liberty Gunboat trainee instructor", or even "Totally Not Klaes Ashford From The Expanse", or "Italian person". These are all things I can understand, and I make an estimated guess over how they would feel in some particular situations, and thus, I can get into character easily.
What I can recommend is thinking about what your character thinks and feels - be descriptive! Make your character yap a lot, make your character show off what they are like and what they've gone through and how they circumvent their problems. Just don't be scared, and be confident in what you're doing. Let the world know about your character, just don't forget about where you started from!