He does not pray and he lives with his girlfriend; he wants to repent and marry her
Question

There is a French Muslim man who does not pray or fast, and he lives with his Christian girlfriend. He wants to repent and fast, but he is using the fact that this woman is living with him as an excuse. Is it permissible for him to marry her now, knowing that tomorrow is the first day of Ramadaan? If that is permissible, then what is the prescribed shar’i procedure for that?.
Praise be to Allah.This person and others should note that not praying is kufr which puts a person beyond the pale of Islam. Islam does not accept for any of its followers not to pray or fast and to live with a girlfriend. 
What you have to do is to advise him and explain to him what Islam really is, which is submitting to the rulings of sharee’ah. The Muslim should be an example to others, especially in that (non-Muslim) country, for he is not representing himself only, rather he is representing Islam which he has embraced and committed himself to. So he has to give up the sin that he is committing and adhere to the rulings of sharee’ah, especially prayer which is the thing that distinguishes between Islam and kufr.
 Secondly: 
We are very happy to learn that he wants to repent, so what is stopping him from repenting? Allaah rejoices over the repentance of His believing slave. When a person turns to Allaah, Allaah turns to him and forgives his sin. So he should hasten to repent and not delay it or make it conditional upon certain things happening, lest he dies without having repented and he meets his Lord with a burden of sin and may even meet Him in state of kufr. 
Tell him that Allaah turns bad deeds into good deeds for the one who repents. Allaah says (interpretation of the meaning): 
“Except those who repent and believe (in Islamic Monotheism), and do righteous deeds; for those, Allaah will change their sins into good deeds, and Allaah is Oft‑Forgiving, Most Merciful”
[al-Furqaan 25:70]
So he must hasten to repent by giving up all that he is doing that incurs the wrath of Allaah, namely not praying and living with his girlfriend. 
See the answers to questions no. 624, 13990, 34905, 22912. 
Thirdly: 
If he repents to Allaah, he should note that it is not permissible for him to marry that girlfriend, not because she is Christian, but because she is a zaaniyah (fornicatress) – according to what he has said. One of the conditions of marrying a woman of the People of the Book (i.e., a Jewish or Christian woman) is that she should be chaste, not a fornicatress or a woman who has a lover or boyfriend. Allaah says (interpretation of the meaning): 
“Made lawful to you this day are At‑Tayyibaat [all kinds of Halaal (lawful) foods, which Allaah has made lawful (meat of slaughtered eatable animals, milk products, fats, vegetables and fruits)]. The food (slaughtered cattle, eatable animals) of the people of the Scripture (Jews and Christians) is lawful to you and yours is lawful to them. (Lawful to you in marriage) are chaste women from the believers and chaste women from those who were given the Scripture (Jews and Christians) before your time when you have given their due Mahr (bridal-money given by the husband to his wife at the time of marriage), desiring chastity (i.e. taking them in legal wedlock) not committing illegal sexual intercourse, nor taking them as girlfriends”
[al-Maa’idah 5:5]
Allaah has stipulated that in order to marry them, they must be chaste. It is not permissible for a Muslim to marry a woman of the people of the Book who is not chaste. Even if a woman is a Muslim but she is not chaste, it is not permissible for a chaste Muslim man to marry her. Allaah says (interpretation of the meaning): 
“The adulterer — fornicator marries not but an adulteress — fornicatress or a Mushrikah; and the adulteress –fornicatress, none marries her except an adulterer — fornicater or a Mushrik. Such a thing is forbidden to the believers (of Islamic Monotheism)”
[al-Nur 24:3]
For more details on this issue, please see the answer to question no. 11195 and 2527. 
If he wants to marry her, that can only happen after he repents and comes back to Islam by starting to pray, and after they both repent from zina (fornication). 
This applies if he wants to marry her. 
The duty of being sincere (naseehah) compels us to tell him of the best way we know of setting straight both his religious and worldly affairs, which is that he should repent to Allaah sincerely, and hasten to leave this woman without any hesitation or delay, and look for another woman who is a chaste, believing Muslim. If he repents to Allaah, he will need someone who understands her religion and will stand by him and encourage him to obey the Most Merciful after this deviation. As for that woman, even if she repents from fornication she will never be able to help him to obey Allaah and she will never be trustworthy with regard to his household, his wealth and his honour, and she will never be fit to raise his sons and daughters. We mean nothing but good by offering this sincere advice. Let him use his mind and avoid being emotional, then he will know that this is right. 
If he looks around him, he will see other Muslims who have married non-Muslim women, and how bad their situations are, and how they regret it and wish that they had not married non-Muslim woman. 
See also the answer to questions no. 20227 and 45645. 
And Allaah is the Source of strength.