It's been a few years, but I've spent time on the boards in high school, college, and the community theatre areas.
The director's style mostly influences how characters are brought about. Some of them are micromanagers and say "I want the character to have this motivation and have this idiosyncratic behavior and etc" and others just say "Hey, use your brain and figure out what the character is doing, I'm just here to tell you where to stand". So even the same show can be interpreted vastly differently by different directors and casts.
I did have one director who forbid us from looking up any motion picture or filmed performances of a show we were doing, and then were allowed to watch the other versions after our show ended. It was a fascinating exercise into our own psyches of how our perceptions worked (example: 'Hello, Dolly!' ... Dolly Levi is a gregarious, energetic woman who is larger-than-life - as portrayed by Barbra Striesand, Carol Channing, and Pearl Bailey; our Dolly played her as a calculating gold-digger because she's a cynic).
As for my own approach, there is an immersion session I do with the period pieces - I listen to the music, read the literature, wear the clothes, and study the history leading up to that era, so I get a feel of where the character is coming from and what outside influences shaped them. I even write in character (which is funny as hell when doing Shakespeare and I've got term papers due). Then there's the 'inspiration pile' in which I find visual aids that remind me of the character - like their likes, favorite things, how they look. I knew one other guy who did this and he went a step beyond by taping all of those pictures to his dressing room mirror (I was a costumer at the time we met, so he never knew of my method).
I know that when it stops being fun, that's when to take a break.
The two best things I had learned were hung up over the doorways of one theatre I worked at:
Entrance: "Please leave your ego outside, there's no room for it in here."
Exit: "Please leave your character inside, there's no room for it out there."