Praise be to Allah.
Allah created men and women and created in each of them attraction towards the other as part of their nature, which no one can deny. Islam pays attention to that and regulates the relationship between the two, which is restricted to marriage; Islam forbids any other kind of relationship which is unacceptable to sound human nature and proper thinking. Woman is not a cheap product for anyone who wants to have his way with her; rather she is respected and should be for one man to take care of her, spend on her and give her her rights. The way to take a woman as a wife is by seeing her first, then asking her guardian for her hand, then giving some wealth to be a dowry for her. All of that means that Islam regards woman as a respectable being, not something that is easily obtained, so that she will not be treated carelessly and looked down upon. What we see in the world of promiscuity is that a man tries to get a woman he likes in order to have his way with her, then he throws her aside so that he can look for another who is more beautiful than her, and so on; and he moves from one prey to another, one girlfriend to another. Thus no family is formed and no children are born; as a result no family is formed on a basis of peace and tranquillity. All of that is rejected in Islam which forbids it in a way that leaves no room for differences of opinion. Every Muslim who goes against that is following his whims and desires and is imitating deviant and immoral people. Islam disavows his conduct and actions.
Based on that, relationships that occur between men and women outside the framework of legitimate, documented marriage are haraam (forbidden) relationships which it is not permissible for a Muslim to form with any woman, whether she is Muslim or non-Muslim. We admire your telling this Muslim man who is going against his religion that if he is looking for sex he should look somewhere else. This confirms what we have said, that woman is not a cheap product. But we do not approve of your going out with him and what happened between you of being alone together and kissing. If you think that this is something normal in your view, that is not the case for us. What he has done are things that are forbidden in Islam and it is not appropriate for a Muslim to do them. If you like each other, then the only way for meetings between you to be permissible is marriage and nothing else. Any meeting or relationship with a woman who is not related to the man that occurs outside the framework of marriage is haraam (forbidden) and is deserving of punishment in the Hereafter.
See, for example, the answer to question no. 34841.
You should note that Islam permits marriage to a Christian woman on condition that she should be a believer in her religion and she should not be an atheist or a person with no religion, and on condition that she should be chaste and honourable, not a fornicator or a woman who has boyfriends. In order for the marriage contract to be valid, you should have a guardian who will give your hand in marriage and there should be two Muslim witnesses to the marriage contract; announcement of the marriage may take the place of witnesses. For more details on this, please see the answer to questions no. 2527 and 12283.
The fact that a woman has children from a previous marriage does not affect her marriage to a Muslim. If someone says that it is not permissible for a Christian or Jewish woman to marry a Muslim unless she is a virgin, that view is not correct. Rather the condition for her being permissible (as a wife for a Muslim man) is that she should be chaste and avoid fornication. This condition also applies to Muslim women; it is not permissible for a chaste Muslim man to marry a woman who commits fornication unless she repents sincerely. And it is not permissible for a chaste woman to marry a man who commits fornication unless he repents sincerely, as we have explained in the answer to question no. 85335.
As the relationship between you has not gone as far as fornication or adultery as you say, praise be to Allah, then there is no reason why he should not marry you, as we have explained in the answer to question no. 148528.
With regard to the Muslim man’s family, they may accept their son marrying a Christian woman who has children. If they refuse, that is not because it is something that is forbidden in Islam; rather it may be for cultural reasons, or perhaps because they think that their son is not able to fulfil the rights of that wife and take care of her children. But they cannot accept their son having a girlfriend with whom he spends time alone or travels with her; if they accept that, then they are undoubtedly going against Islamic teachings.
The issue of whether or not his family will accept that is a social issue, not a religious ruling. It is not one of the conditions of a man’s marriage that his family should accept it, although that is something that is desirable.
Even though the man is Muslim – as you say – that does not prevent us from speaking the truth with regard to what he has done with you and the ruling on his relationship with you. We would give you some further advice: do not pay any attention to the sweet talk from anyone who has a relationship with you outside the framework of marriage. That includes this man’s saying that he has become “a captive love” to you and that he “will never exploit” you, and other such enchanting words. All of that is most likely aimed at getting what he wants, which is to have his way with you, then he will go off and look for someone else. What we are saying is only based on understanding the reality of illicit relationships, especially those that begin in chat rooms and on Facebook. We hope that you will not be deceived by this sweet talk. If this man is sincere in his love for you, then let him prove it by marrying you. As for proving that by going out together to watch a movie or to a restaurant, none of that is proof of sincerity of feelings, so do not pin your hopes on it. Even if non-Muslims do that and women accept it, Islam forbids it and does not approve it, and it warns women against being fooled by it.
Now the opportunity presents itself for us to tell you of something that will be better for you in this world and in the Hereafter, which is to enter Islam and become a Muslim woman who affirms the Oneness of Allah, the Creator of the universe and the Provider of all creation. This religion to which we are calling you is the religion of Jesus (peace be upon him) and the religion of Abraham and their fellow Prophets and Messengers, all of whom worshipped One Lord, namely Allah, may He be glorified and exalted. They devoted their worship to Him alone, with no partner or associate, and they called the people to do likewise. Islam is the religion that brought rulings which are suited to all times and all places; it addresses the interests of individuals and societies. You will feel great happiness, as has been experienced by those who have gone ahead of you on this path. After that, Allah will make it easy for you to find a suitable husband who will take care of you, give you your rights and seek reward with Allah by taking care of your children, and perhaps you and he will attain the reward of their becoming Muslim and you will be together with them in this world and the Hereafter. If you do not become Muslim, then you must understand that even if you are together with a Muslim man as his wife in this world, in the Hereafter you will inevitably be separated from him. This underlines what we have said, that your becoming Muslim means attaining success in this world and in the Hereafter. Being together with your husband and children in the Paradise of eternity is a great success for which wise people strive and those who are doomed miss out on.
We ask Allah to guide you to that which is best for you in this world and in the Hereafter, and to make it easy for you to find a righteous and good husband.
And Allah knows best.