Offering Condolence Message to Non-Muslims
Question

Is it permissible to offer condolences to Christians, and if so, how is that done?
Praise be to Allah.Offering Condolence Message to Non-Muslims
Yes, it is permissible to offer non-Muslims condolences at times of bereavement, to visit them when they are sick, and to console them when calamity strikes. 
Anas (may Allah be pleased with him) said: “There was a Jewish boy who used to serve the Prophet (peace and blessings of Allah be upon him). He fell sick, and the Prophet (peace and blessings of Allah be upon him) came to visit him. He sat at his head and said, ‘Become Muslim.’ (The boy) looked at his father who was with him, and he (the father) said to him, ‘Obey Abu’l-Qasim ( (peace and blessings of Allah be upon him)).’ So he became Muslim. Then the Prophet (peace and blessings of Allah be upon him) left, saying, ‘Praise be to Allah who has saved him from the Fire.’” (Narrated by al-Bukhari, 1356) 
Anas reported that a Jew invited the Prophet (peace and blessings of Allah be upon him) to eat barley bread and other food, and he accepted the invitation. (Narrated by Ahmad, 13201, with a sahih isnad)
How to offer condolence to non-Muslims
If a Muslim does that, he should do so with the intention of calling them to Islam and softening their hearts towards Islam; he should call them in an appropriate manner and at an appropriate time. 
Whilst consoling non-Muslims, a Muslim should not pray for forgiveness or mercy or Paradise for their dead, because Allah says (interpretation of the meaning): 
“It is not (proper) for the Prophet and those who believe to ask Allah’s forgiveness for the Mushrikun, even though they are of kin.” [al-Tawbah 9:113]
Rather he should pray for them in a way that is appropriate, encourage them to be patient and steadfast, and remind them that this is the way of Allah with His creation. 
For more about condolences, please see these answers: 12326 , 12530 , 174755 , and 215016 . 
And Allah knows best.