She separated from her first husband by means of khul‘ without his consent and married another man, then she went back to her first husband

Praise be to Allah.

The khul‘ that you did yourself is invalid and has no effect, because the woman does not have the right to annul the marriage contract, either by means of khul‘ or talaaq (divorce), just as she does not have the right to do the marriage contract. Rather the woman may ask to separate from her husband by means of khul‘ if there is good reason to do so, and if he accepts then he is the one who separates from her by means of khul‘ or talaaq. If he does not agree then she should refer the case to the shar‘i judge (qaadi) who will either force him to give a khul‘ or talaaq, or not. 

If the woman marries someone other than her first husband before having separated from him in a valid manner, either through talaaq or annulment of the marriage, or the husband’s death, then that (second) marriage contract is invalid, according to scholarly consensus. If she was aware that it was invalid then she is a zaaniyah (adulteress) who is subject to the hadd punishment, and so is he (the second husband). If she was not aware of that because she thought that she had the right to end the first marriage herself by means of khul‘, and that that could be valid, then she is excused because of her ignorance and the hadd punishment is not to be carried out on her, but her second marriage is also invalid and she has to leave him and observe ‘iddah, then go back to the first husband. 

Ibn Qudaamah (may Allah have mercy on him) said: 

With regard to invalid marriages, such as the marriage of a married woman or one who is observing ‘iddah and the like, if they were aware of what is halaal and what is haraam, then they are both guilty of zina and are thus subject to the hadd punishment, and any child born as a result is not to be attributed to the man. … and ‘iddah becomes obligatory once she has been alone with him; if he dies, she must ‘iddah and mourning. All of that is required as a precautionary measure for her sake.

End quote from al-Mughni, 7/13 

It says in al-Mawsoo‘ah al-Fiqhiyyah, 8/123-124 

The fuqaha’ are unanimously agreed that it is obligatory to observe ‘iddah and any child born should be attributed to the man in the case of a marriage concerning which the madhhabs differ, once the marriage has been consummated, such as a marriage without witnesses or without a guardian, or marriage of a pilgrim in ihram during Hajj, or shighaar (quid-pro-quo marriage). The Hanbalis added that the same applies once it is proven that they have been alone together. 

They are also agreed that ‘iddah is required and the child is to be attributed to the man in the case of a marriage concerning which there is consensus that it is invalid once the marriage has been consummated, such as the marriage of any woman during her ‘iddah, the wife of another man and mahrams, if there is some doubt which would cause the hadd punishment to be waived, such as if the parties involved did not know that it is haraam, and because the basic principle according to the fuqaha’ is that in any case of marriage where the hadd punishment is to be waived, the child is to be attributed to the man. 

But if there is no doubt that would cause the hadd punishment to be waived, such as if the parties involved were aware that it is haraam, then the child is not to be attributed to the man according to the majority and also according to some of the Hanafi shaykhs, because when the hadd punishment becomes obligatory, attribution of the child to the man is not proven. And according to Abu Haneefah and some of the Hanafi shaykhs, attribution is proven because there is some doubt about the invalidity of the marriage contract.  

See also al-Mawsoo‘ah al-Fiqhiyyah, 29/339 

See also the answer to question no. 171791 

To sum up: 

In all cases the woman does not have the right to separate from her husband by means of khul‘ by herself, as happened in your case. 

Based on that, your second marriage to this liar and cheater is invalid and does not count. You have to observe ‘iddah following your separation from him, as in the case of a woman who is divorced by talaaq. 

Some of the scholars are of the view that in this case to prove that you are not pregnant from him, one menstrual cycle is sufficient. 

See ash-Sharh al-Mumti‘, 13/381-383 

And Allah knows best.