What is the ruling on asking a Christian to help me obtain a job?
Is it permissible for me to ask after him and greet him?

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 is His slave and Messenger.

It is permissible for a Muslim to ask for the help of a disbeliever who is at peace with Muslims for non-religious matters like medicine, agriculture, industry, and the like. Employment is considered one of these matters. The Prophet, sallallaahu ‘alayhi wa sallam , hired 'Abdullaah Ibn 'Urayqit, who was a polytheist at the time, to lead him on his way to Madeenah during his Hijrah (emigration).  [Al-Haakim in Al-Mustadrak and At-Tabraani in Al-Mu‘jam Al-Kabeer]

Also, it is permissible for a Muslim to accept a gift from a disbeliever and to offer him gifts as well. The Prophet, sallallaahu ‘alayhi wa sallam, accepted gifts from disbelieving kings, and Allaah The Almighty Says (what means): {Allaah does not forbid you from those who do not fight you because of religion and do not expel you from your homes - from being righteous toward them and acting justly toward them. Indeed, Allah loves those who act justly.} [Quran 60:8] 

This relationship requires an exchange of interests, which is different from forbidden alliances with disbelievers, because forbidden alliances with them means all or any of the following:

-        Allying with them against Muslims;

-        Being indifferent to the fact that they are polytheists;

-        Copying their lifestyle and loving them. 

With regards to greeting them, some scholars consider it  permissible, like Sufyaan Ibn ‘Uyaynah .  He grounded his  opinion on the Quranic verses (which mean):

•        {Allaah does not forbid you from those who do not fight you because of religion and do not expel you from your homes - from being righteous toward them and acting justly toward them. Indeed, Allaah loves those who act justly.} [Quran 60:8]

•        {There has already been for you an excellent pattern in Abraham and those with him.} [Quran 60:4]

•        Ibraaheem (Abraham) greeted his father saying {"Peace will be upon you.} [Quran 19:47]

Al-Qurtubi, the Maaliki scholar interpreted the Hadeeth where the Prophet, sallallaahu ‘alayhi wa sallam, commanded Muslims not to be first to greet Jews and Christians to only apply when there is no need for a greeting.

This opinion is also supported by a Hadeeth narrated on the authority of Usaamah Ibn Zayd .  In this Hadeeth it was narrated that the Prophet, sallallaahu ‘alayhi wa sallam, passed by a gathering where there were Muslims, disbelievers and Jews, and greeted them and then stopped to invite them to Islam. This Hadeeth proves that it is permissible for a Muslim to greet disbelievers if there is a need.

Also, 'Abdulaah Ibn Mas‘ood greeted a disbeliever who was traveling with him. When ‘Alqamah reminded him about the prohibition of greeting a disbeliever first, Ibn Mas‘ood told him that this was the right of company on the road. [Al-Bayhaqi in Shu‘ab Al-'Eemaan] Consequently, it is permissible to greet a disbeliever when there is a need, or for the right of company.

With regards to asking after him, then on face value this is permissible, as it was narrated that the Prophet, sallallaahu ‘alayhi wa sallam, visited a sick Jewish boy [Al-Bukhari, Abu Daawood and Ahmad in Al-Musnad]  This Hadeeth indicates the permissibility of visiting a sick disbeliever which logically denotes the permissibility of simply asking after him.

Allaah Knows best.