According to where one live, to change one's name - if it's not really required - might not be the best idea. For instance using a Muslim name tend to get you through hardship in many situations, such as looking for a new job, in the airport and so on. We should not make it hard upon ourselves:
And Allah wants to lighten for you [your difficulties]; and mankind was created weak.
Do you live in a muslim or Arabic speaking country, then it might be much more accepted and "normal" with changing to an Arabic name.
Now, the name one has is personal and is given by one's parents, how glad would they be if you changed your name? How will this affect your relationship with your parents or family? Might it lead to difficulties or even splitting one's ties with one's family?
Maintaining the bonds of kinship (silatur-rahim) indeed enjoys extraordinary importance in Islam. Conversely, severing the ties (qata-ur-rahim) is very high on the list of enormities. On two occasions, Allah cursed the one severing family ties: “And those who break the covenant of Allah, after its ratification, and sever that which Allah has commanded to be joined (i.e. they sever the bond of kinship and are not good to their relatives) and work mischief in the land, on them is the curse, and for them is the unhappy home (i.e. Hell)” (Qur’an, 13:25)
Allah also says: “So would you perhaps, if you turned away, cause corruption on earth and sever your [ties of] relationship? Would ye then, if ye were given the command, work corruption in the land and sever your ties of kinship? (Qur’an, 47:22-23)
Now the name Christine isn't a name that must be changed.
The name Christine is of French and English origin, meaning "follower of Christ" Reference can be found in Wikipedia, urbandictionary.com and other sources.
Christ (also masih in arabic, mentioned in Quran several times) means the chosen or anointed. Check it up.
So the follower of the anointed, isn't that a beautiful name?
If we can name our sons for the chosen (Mustafa, Mokhtar) in Arabic, why can't someone be called the follower of the chosen in another language?
My conclusion is that this name is not haram.