When the servant of Allah turns to his Lord in the middle of the dark night, and whispers to Him, Allah establishes His light in his heart . . . then He tells the angels: O my angels, look at my servant. He has turned to Me in the middle of the dark night while the false ones are playing, and the heedless ones are sleeping; bear witness that I have forgiven him. - Holy Prophet (s)
There is no good deed except that its reward has been outlined in the Qur’an, except the Salatul Layl. Almighty Allah has not specified its reward due to its greatness with Him. He says (about the reward of those who recite Salatul Layl): So no soul knows what is hidden for them of that which will refresh the eyes; a reward for what they did.(32:17) - Holy Prophet (s)
Many people may try to pray the night prayer, tahajjud, or salatul layl, more often during the month of Ramadan.
Here is a good lecture about 15 minutes long that you can watch or listen to which explains how to perform salatul layl. The actual English lecture begins at 1:30.
And here is a nice book about salatul layl including dua'as, etc. You could print out pages from it and hold while praying if that would be helpful to you.
Here also is another resource on salatul layl including a concise method of performing it.
Here is the concise method reported by Allamah Majlisi in the above source:
Salat al-Lail is 11 Rak`at in which:
1. The first 8 Rak`at are prayed in four Salat of two Rak`at each with the intention of Salat al-Lail (after each two Rak`at, give the Salam, finish the Salat and then stand for the next set of Salat.)
2. Following this, two Rak`at would be read with the intention of Salat ash-Shaf` (this Salat is performed exactly as Salatul Fajr).
3. Once this is recited, one Rak`at with the intention of Salat al-Witr would be recited. In this Salat, recite Surah Al-Fatiha once, Surah al-Ikhlas three times followed by Surah al-Falaq and Surah al-Nas once each.
Once this part of the Salat is finished, we raise our hands in Qunut such that our palms are facing the sky and our hands are level with our eyes and perform the following.
1. Recite the following 70 times:
أَسْتَغْفِرُ اللٌّهَ رَبِّي وَأَتُوبُ إِلَيْهِ
“I seek repentance from Allah, my Lord and to Him I turn back.”
2. Followed by reciting the following seven times:
أَسْتَغْفِرُ اللٌّهَ الَّذِي لاٌ إِلٌهَ إِلاٌّ هُوَ الْحَيُّ الْقَيُّومُ وَأَتُوبُ إِلَيهِ
“I seek repentance from Allah the One whom there is no god except Him, the Living and Self-Subsisting and to Him I turn back.”
3. One of the most important acts in the Qunut of the Salat al-Witr is supplication for forty believers (men or women) amongst one’s family, friends, those who have asked to be prayed for, the deceased and the Shi`a of Amir Al-Mo’minin `Ali ibn Abi Talib (peace be upon him). Whosoever does this will be assured that his supplications are answered.
4. One should then say the following three hundred times:
“I ask for forgiveness.”
5. Following this, the following should be said once:
رَبَّ اغْفِرْلِي وَارْحَمْنِي وَتُبْ عَلَيِّ إِنَّكَ أَنْتَ التَّوٌابُ الرَّحِيمِ
“Lord, please forgive me and have mercy upon me and turn back towards me. Verily You are the Oft-Turning back, Most Merciful”
May Allah (Glory and Greatness be to Him) hasten the advent of the 12th Imam (may Allah hasten his return) to fill this world with justice and equality.