En plus means furthermore here.
Déjà je suis née fille, je vais pas en plus venir parler la première.
might be rewritten to:
Déjà, je suis née fille. Je ne vais pas, en plus, venir parler la première.
which might be translated by:
For a start, I was born a girl. I'm not going to take the floor first on top of that.
The first part is unusual. One would expect Déjà, je suis une fille unless the sentence is from a transgender person.
The second part probably means several people are going to speak on stage or something, and the person "born a girl" doesn't want to be the first one to take the floor, precisely because of being born a girl. The context should explain why (s)he thinks it is a reason not to.