What is the ruling on a Christian who makes public his embracing of Islam in order to marry a Muslim woman? What are the conditions that he must fulfill to be considered a real Muslim?

All perfect praise be to Allaah, the Lord of the worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and messenger.

The scholars agreed in a consensus  that it is not permissible and not valid for a non-Muslim –whether a Jew or Christian or else -to marry a Muslim woman, as Allaah Says (what means): {And do not marry polytheistic women until they believe. And a believing slave woman is better than a polytheist, even though she might please you. And do not marry polytheistic men [to your women] until they believe. And a believing slave is better than a polytheist, even though he might please you. Those invite [you] to the Fire, but Allaah invites to Paradise and to forgiveness, by His permission. And He makes clear His verses to the people that perhaps they may remember.}[Quran 2:221]

However, if the non-Muslim embraces Islam and proves to be a good Muslim and has good morals, then it becomes permissible for him to marry a Muslim woman because he is a Muslim in this case.

With regard to the conditions that must be fulfilled for us to consider this person a Muslim, then he should utter the Two Testimonies of Faith. By doing so, he becomes a Muslim, then he should be committed to performing the religious obligations without which Islam is void, like performing the prayer, and renouncing actions that takes him out of Islam, like wearing the cross. Once these conditions are met, we consider him to be a Muslim, and we leave what is in his heart to Allaah.

‘Umar mentioned that people can only judge a person’s external acts, and only Allaah can judge what is in their heart.

Allaah Knows best.