I'm thinking of pursuing a relationship with a married woman that isn't happy with her situation
Have you thought about this a lot? How do you see it working out?
I will get to be a part of a happiness I did not have
You can't help who you're attracted to, and want her to be happier. But it's her marriage and you shouldn't risk it to make yourself happy.
I just have to be patient and keep my nose clean.
Yes, if things don't work out between them then you could go ask her out. For now though, you should let things happen naturally and just wait-- it is better for you both in the long run to not push it.