While the only characters I have the good fortune to write have been ones that are already part of their own worlds, I still like listening to music from their show, game or film to get me in the right mindset. Then I just take a walk, listening to it until an idea pops into my head.
My main way of trying to get to know a characters it finding music from their material that they would dance, run or fight to. It sounds weird, but it just helps relate to the character more if I have a theme I can see them doing a lot of actions in. Of course, I may actually use their battle themes if they have one, but that hasn't come up yet
But music does help, and I do spend most of my time brainstorming situations rather than actually writing them
"Kindeys. I've got new kidneys! I don't like the colour."
- The 12th Doctor
"We have 80 Million ancestors; one of them has got to be Winnie the Pooh."
- David Mitchell