Chatting Before Marriage in Islam
Question

I have never spoken to her, and generally do not speak to women. We sometimes exchange Salams.
How do I propose to her and approach her for marriage, since I am a practising Muslim, and do not talk to sisters, what is the best way?
Should I go and speak to her and get to know her first, without stepping beyond the boundaries of Allah? or shall I propose straight away?
I am afraid that because she does not really know me, and that we are from different cultural backgrounds I will be rejected instantly if I propose straight away without getting to know her first.
Whilst on the other hand I fear that I am doing something Un-islamic if I talk to her and get to know her.
I am in a difficult situation. What is the best thing to do?
Praise be to Allah.Etiquette of talking to non-Mahram women
It is permissible for a man to speak to a non-Mahram woman , subject to important guidelines and conditions, the purpose of which is to prevent Fitnah (temptation) and sin. These conditions include: 

It should be done without being alone with her.
It should not go beyond permissible topics.
There should be no Fitnah. If his desire is stirred by talking to her or if he starts to enjoy it, then it is prohibited.
The woman should not speak in a soft manner,
The woman should be wearing full Hijab and be modest, or he should speak to her from behind a door. It is better if they speak on the phone, and even better if they communicate via letters or email , for example.
That should not go beyond what is necessary. 

If these conditions are met and there is no fear of Fitnah, then it is allowed, and Allah knows best. 
Shaykh Salih Al-Fawzan (may  Allah preserve him) said, in his answer on the ruling about young men speaking to young women on the telephone:
“It is not permissible for young men to speak to young women because of the Fitnah involved, unless the girl is engaged to the man who is talking to her , and they talk only about matters pertaining to their engagement; but it is preferable and safer for him to speak to her guardian about that.” (Al-Muntaqa min Fatawa Ash-Shaykh Salih Al-Fawzan, 3/163, 164)
Chatting before Marriage in Islam
You have not proposed to this girl yet, so you have to be very careful and avoid exposing yourself to the causes of Fitnah by taking all possible precautions to achieve your goal without approaching this girl. 
The basis for this is two verses from the Book of Allah: 
1 – “O wives of the Prophet! You are not like any other women. If you keep your duty (to Allah), then be not soft in speech, lest he in whose heart is a disease (of hypocrisy, or evil desire for adultery) should be moved with desire, but speak in an honourable manner.” [Al-Ahzab 33:32 – interpretation of the meaning] 
2 – “And when you ask (his wives) for anything you want, ask them from behind a screen, that is purer for your hearts and for their hearts” [Al-Ahzab 33:53 – interpretation of the meaning] 
What to Look for in a Spouse in Islam
Finally I would like to remind you that the Muslim’s standards when choosing a wife should be the standards encouraged by the Messenger of Allah (peace and blessings of Allah be upon him) when he said: “Choose the one who is religiously-committed , may your hands be rubbed with dust [i.e., may you prosper].” (Narrated by Al-Bukhari, 5090 and Muslim, 1466) 
Limits of relationship with a fiancé
 I also warn you against everything that may lead you into doing forbidden things or bring you close to that, such as being alone with her , going out with her, etc. I ask Allah to make it easy for you to find a woman who will help you to obey Him.
And Allah knows best.