Salamu Aleicum Rinoa...
I have converted to Islam myself, and although not all of my family knows (so in a way I am in the same position as you are), I do have some advise given to me by other Muslims:
1- First of all, keep in mind at all times that you are bringing your family excellent news by telling them you are a Muslim. Do not fear, in reality this is a good thing.
2- Think of all the wonderful reasons that made you turn into a Muslim. Tell your family about this when you tell them.
Also, take the person of your family that is closest to you and tell him/her first. See his reaction. This will be an indication of how the other ones may react.
If you feel you are at threat when telling, do not do it. Other Muslims, when our religion was first born, also hid it for a while to preserve their lives.
Lastly, may Allah guide you whilst you are braking the good news.