She became Muslim but her family did not; should she honour her family even though they mistreat her?
Question

I became Muslim when I was young. My father threw me out of the house so I moved to a Muslim country with my husband to live there. I still keep in touch with my mother. Is there any sin on me for not keeping in touch with my father, knowing that he tells people I am a prostitute and he is threatening to kill me and my family?.
Praise be to Allah.We praise Allah for having guided you to Islam. Undoubtedly Allah has blessed you greatly by choosing you from among your family to be the first of them to enter this religion. We ask Allah to make you the cause of them also entering Islam. 
What you have done of calling your family to Islam is what Allah has enjoined upon you; they take priority over others when it comes to da’wah and telling them the truth. 
Allah says (interpretation of the meaning): 
“Say (O Muhammad): ‘This is my way; I invite unto Allah (i.e. to the Oneness of Allah — Islamic Monotheism) with sure knowledge, I and whosoever follows me (also must invite others to Allah, i.e. to the Oneness of Allah — Islamic Monotheism with sure knowledge). And Glorified and Exalted be Allah (above all that they associate as partners with Him). And I am not of the Mushrikoon (polytheists, pagans, idolaters and disbelievers in the Oneness of Allah; those who worship others along with Allah or set up rivals or partners to Allah)’”
[Yoosuf 12:108]
“And warn your tribe (O Muhammad) of near kindred”
[al-Shu’ara’ 26:214]
The person who calls others to Allah should be gentle and subtle in his approach, especially with his family. Allah has commanded us to treat our parents kindly, even if they are kaafirs and call us to kufr. Allah says (interpretation of the meaning): 
“Invite (mankind, O Muhammad) to the way of your Lord (i.e. Islam) with wisdom (i.e. with the Divine Revelation and the Qur’aan) and fair preaching, and argue with them in a way that is better. Truly, your Lord knows best who has gone astray from His path, and He is the Best Aware of those who are guided”
[al-Nahl 16:125] 
“And We have enjoined on man to be good and dutiful to his parents; but if they strive to make you join with Me (in worship) anything (as a partner) of which you have no knowledge, then obey them not. Unto Me is your return and I shall tell you what you used to do”
[al-‘Ankaboot 29:8]
If any of them does not respond, his misguidance is his own problem and Allah will not make the daa’iyah accountable for any of his sin. 
Allah says (interpretation of the meaning): 
“Whoever goes right, then he goes right only for the benefit of his ownself. And whoever goes astray, then he goes astray to his own loss. No one laden with burdens can bear another’s burden. And We never punish until We have sent a Messenger (to give warning)”
[al-Isra’ 17:15]
What you did, migrating to one of the Muslim countries and getting married, was the right thing to do. The Muslim cannot usually maintain his or her religious commitment in a hostile environment where he or she is a stranger; it is especially hard for a woman who has no power and no strength except with Allah. This difficulty is demonstrated by what your father did, throwing you out of the house when he found out that you are a Muslim, 
Your keeping in touch with your mother and asking after your father is something for which you deserve praise; this is something that Allah has enjoined upon you. The rights of parents are great, so do not cut off your ties with them, even if they mistreat you. Try to get in touch with your father and speak kindly to him; perhaps that will be a cause of his being guided and will dispel the hardness in his heart towards you. 
With regard to your father’s threats, do not pay any attention to them and do not worry about them; nothing will happen to you or your husband or your family except that which Allah has decreed for you. So take precautions and seek refuge with Allah, for He is the Best of protectors and He is the Most Merciful of those who show mercy. 
With regard to his slander and accusations against you, this comes under the heading of the kaafir’s persecution of the Muslim. The honour of our Prophet Muhammad (peace and blessings of Allah be upon him) was slandered when his wife and our mother, the Mother of the Believers ‘Aa’ishah (may Allah be pleased with her) was accused of adultery; and it was said that he was a sorcerer, a soothsayer and a madman. Similar accusations were made against his brother Prophets as well. Be patient and trust that Allah will give you a way out and will relieve you of your worry; seek His help and always make du’aa’ to Him and turn to Him, for He is the Best of supporters and helpers. 
We ask Allah to make you steadfast in adhering to His religion and to increase you in guidance, insight and knowledge.