A questioner asks what is Khula and what is the procedure? Mufti Abdur-Rahman ibn Yusuf explains. 

Khula is divorce that is requested by a woman if she feels her husband is not giving her her rights. For example if he hasn’t provided her adequate housing with her own amenities without any interference. If he is not providing her with clothing and food and the basic expenses that she would need to be with him to be a wife plus other complicated issues. 

If the marriage is irreconcilable the wife can tell the husband she wants to get out of the marriage. If the husband is willing to negotiate then she doesn’t need to go to anyone. Some husbands refuse to give a divorce out of oppression and that is wrong. 

If he is refusing then the wife is able to go to the Shariah arbitration panel where she will put your case to them. They will look at everything including the history and then they will issue a divorce. 
This is an excellent breakdown of the procedure of khula.