Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions. Allah has permitted for us to marry the women of the People of the Book provided they are chaste.
Allah Says (interpretation of meaning): {…foods, which Allâh has made lawful (meat of slaughtered eatable animals, etc., milk products, fats, vegetables and fruits, etc.). The food (slaughtered cattle, eatable animals, etc.) 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,…}[5:5].
There is no harm if a priest is present at the ceremony of marriage, and he should be present if he is the guardian of the bride, because it is not permissible for a Muslim to be a guardian of a non-Muslim, the same applies for the person who conducts the marriage.
There is no harm for the priest to conduct the marriage if he is entrusted to do so by the guardian of the bride and the marriage is not valid without this (without the consent of the guardian).
Islam sets some conditions which should be fulfilled for the marriage to be valid (and without these conditions, the marriage is invalid):
1) Defining the bride and bridegroom; it is not permissible for the guardian to say, ‘I agree to marry my daughter to you’ while he has other daughters. The bride has to be designated by name.
2) Agreement of both the bride and bridegroom.
3) The Wali (guardian) should be present, Ahmed and Abu Dawud reported that the Prophet (Sallallahu Alaihi wa Sallam) said: There is no marriage without a Wali.
4) Two witnesses should be present.
5) The bride and bridegroom should have no marriage impediments (i.e. one of them should not be forbidden to the other due to lineage or suckling and they should not be in a state of Ihram – like when performing the rituals of Hajj).
Entering the church for the purpose of conducting the marriage is not permissible for a Muslim, because if he does so, he approves of their disbelief, and helps them in their religion and other very serious matters.
In addition to this, the husband is not permitted to prevent his Christian wife from practising her religion.
On the other hand, you have to know that it is better for a Muslim to marry a Muslim wife. This is more appropriate for his salvation, and the appropriate bringing up of his children.
It is not certain that the Christian wife would fulfill her promise in educating the children according to Islamic principles.