How do you write a strong female character?
For me, a strong female character is a woman who has embraced most, if not all, of the things that make her, well, herself. She is multi-layered, filled with emotional searching, psychological depths, and sexual power (but not necessarily overly sexualized as a character trait). She owns both her failures and her successes. She isn't afraid to exercise all the things that make her who she is. She may possess quiet strength, or she may be a smart-ass quipper who holds her own with the boys (so to speak). She isn't defined by the man (or woman) in her life, nor by the fashions she wears.
She is unapologetically herself.
Now, all that said, in a story she shouldn't necessarily start off fully strong in that sense defined above, because, well, getting her there is what storytelling is for.
(Let it be said too that this same definition can apply to writing strong male characters.)