I want to marry my cousin, but she doesn't really consider herself to be a Muslim and doesn't practice it. She comes from a good muslim family, as do I, and I practice my faith, and I try and educate her. My question is am I OK to marry her? Does it eventually come down to influence and how my children would be brought up?

If she consider herself christian, then you can not marry her as an obligatory precaution, in which you can refer to another marjaa, like Sheikh Ishaq Fayadh, but I do not recommend that you marry her, as this will effect your kids badly.
However, if she consider herself as infidel, then you can not marry her at all.