All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
Before proposing to this girl, we advise to do two things:
Firstly, perform Istikhaarah prayer about this matter. Pray two supererogatory Rak'ahs then make Du'a to Allah to bestow the right guidance upon you. The Istikhaarah supplication is: “O Allah, I consult You as You are All-Knowing and I appeal to You to give me power as You are Omnipotent, I ask You for Your great favor, for You have power and I do not, and You know all of the hidden matters. O Allah! If you know that this matter (then he should mention it) is good for me in my religion, my livelihood, and for my life in the Hereafter, (or he said: 'for my present and future life,') then make it (easy) for me. And if you know that this matter is not good for me in my religion, my livelihood and my life in the Hereafter, (or he said: ‘for my present and future life,’) then keep it away from me and take me away from it and choose what is good for me wherever it is and please me with it.”
Secondly, seek the advice of pious, knowledgeable persons who know such girl. Having done the above, if you feel like marrying her, propose that to her Wali (guardian). You have the right to look at her face and talk to her in the presence of her Wali or some other trustworthy Mahaarim or someone whose presence makes seclusion ineffective. On the other hand, there are no special religious things one should do when proposing to marry.
For more benefit on how to perform Istikhaarah prayer and how to supplicate Allah, please refer to Fataawa 172573, 81434, and 82964.
Allah knows best.