Tahajjud prayer is part of the Sunnah of the Prophet (peace and blessings of Allah be upon him) which Muslims should follow all the time; it’s not something confined to Ramadan.
In his famous work, Fiqh As-Sunnah, Sheikh Sayyid Sabiq, elaborates on the subject as follows:
“TahajjudPrayer: Its excellence and merit, according to the Qur’an and Sunnah
Ordering His Messenger to perform Salatul Tahajjud, Allah Almighty says: “And some part of the night awake for prayer, a largess for thee. It may be that thy Lord will raise thee to a praised position.” (Al-Isra’: 79)
This order, although it was specifically directed to the Prophet, also refers to all Muslims, since the Prophet is a perfect example and guide for us in all matters.
Moreover, performing Tahajjud prayers regularly qualifies one as of the Righteous and earns one Allah’s bounty and mercy. Allah says: “Lo! Those who keep from evil will dwell amid gardens and watersprings, taking that which their Lord gives them. For, lo, they were doers of good. They used to sleep but little of the night and before the dawning of each day would seek forgiveness.” (Adh-Dhariyat:15-18)
In praising those who perform the late night prayers, Allah says: “The slaves of the Beneficent are they who walk upon the earth in humbleness, and when the ignorant address them, they say: ‘Peace’ and they who spend the night prostrating before their Lord and standing.”(Al-Furqan: 63-64)
Next to these Qur’anic verses, there also exist a number of hadiths that reinforce the importance of Tahajjud:

`Abdullah ibn as-Salam reported: “When the Prophet (peace and blessings of Allah be upon him) came to Medinah, the people gathered around him and I was one of them. I looked at his face and understood that it was not the face of a liar. The first words I heard him say were: ‘O people, spread the salutations, feed the people, keep the ties of kinship, and pray during the night while the others sleep and you will enter Paradise in peace.”‘ This is related by al-Hakim, Ibn Majah, and at-Tirmizhi who considers it authentic and sound hadith.
Salman al-Farsi quoted the Prophet (peace and blessings of Allah be upon him) as saying: “Observe the night prayer, it was the practice of the righteous before you and it brings you closer to your Lord and it is penance for evil deeds and erases the sins and repels disease from the body.”
Etiquettes of Tahajjud prayer
The following acts are recommended for one who wishes to perform the Tahajjud prayers:
1- Upon going to sleep, one should make the intention to perform the Tahajjud prayers. Abu ad-Darda’ quoted the Prophet (peace and blessings of Allah be upon him) as saying: “Whoever goes to his bed with the intention of getting up and praying during the night, but, being overcome by sleep, fails to do that, he will have recorded for him what he has intended, and his sleep will be reckoned as a charity (an act of mercy) for him from his Lord.” This is related by an-Nasa’i and ibn Majah through the authentic chain of narrators.
2- On waking up, one should wipe one’s face, use a tooth brush, and look to the sky and make the supplication which has been reported from the Prophet (peace and blessings of Allah be upon him): “O Allah! There is no god but Thee, Glory be to Thee, I seek Thy forgiveness for my sins, and I ask for Your mercy. O Allah, increase my knowledge and let my heart not swerve after You have guided me, and bestow mercy upon me from Thyself. All praise be to Allah who has given us back life after our death and unto Him is the resurrecti
on.”
Then, one should recite the last ten verses of Surat Al `Imran starting with, “Behold! in the creation of the heavens and the earth, and the alternation of night and day,- there are indeed Signs for men of understanding.” Then one should say, “O Allah, to You belongs the praise. You are the Light of the heavens and the earth and what is therein. And to You belongs the praise. You are the truth and Your promise is true. Meeting with You is true; the Paradise is true, the Fire is true. The Prophets are true, and Muhammad is true and the Hour is true ! O Allah, to You have I submitted, in You have I believed, in You have I put my trust, to You have I turned, by You I argue and to You do I turn for my decisions. Forgive me of my former and latter sins, and those done in private and those done in public. You are Allah, there is no god besides Thee.”
3- One should begin Tahajjud with two quick rak`ahs and then one may pray whatever one wishes after that. `A’ishah said: “When the Prophet prayed during the late-night, he would begin his prayers with two quick rak`ahs.” Both of these reports are related by Muslim.
4- One should wake up one’s family for Tahajjud. Abu Hurairah quoted the Prophet (peace and blessings be upon him), as saying: “May Allah bless the man who gets up during the night to pray and wakes up his wife and who, if she refuses to get up, sprinkles water on her face. And may Allah bless the woman who gets up during the night to pray and wakes up her husband and who, if he refuses, sprinkles water on his face.”The Prophet (peace and blessings of Allah be upon him) also said: “If a man wakes his wife and prays during the night or they pray two rak`ahs together, they will be recorded among those (men and women) who (constantly) make remembrance of Allah.” This is related by Abu Dawud and others through an authentic chain of narrators.
5-If one gets sleepy while performing Tahajjud one should sleep. This is due to the hadith narrated by `A’ishah (may Allah be pleased with her) as she quoted Allah’s Messenger as saying: “When one of you gets up during the night for prayer and his Qur’anic recital gets mixed up to the extent that he does not know what he says, he should lie down.”This is related by Muslim.
Recommended time for Tahajjud
Tahajjud may be performed in the early part of the night, the middle part of the night, or the latter part of the night, but after the obligatory salatul `Isha’ (Evening prayer).
While describing the Prophet’s way of performing prayer, Anas said: “If we wanted to see him praying during the night, we could see him praying. If we wanted to see him sleeping during the night, we could see him sleeping. And sometimes he would fast for so many days that we thought he would not leave fasting throughout that month. And sometimes he would not fast (for so many days) that we thought he would not fast during that month.” This is related by Ahmad, al-Bukhari, and an-Nasa’i.
Commenting on this subject, Ibn Hajar says: “There was no specific time in which the Prophet (peace and blessings be upon him) would perform his late night prayer; but he used to do whatever was easiest for him.”
Best time for Tahajjud
It is best to delay this prayer to the last third portion of the night. Abu Hurairah quoted that the Messenger of Allah, as saying: “Our Lord descends to the lowest heaven during the last third of the night, inquiring: ‘Who will call on Me so that I may respond to him? Who is asking something of Me so I may give it to him? Who is asking for My forgiveness so I may forgive him?”‘This is related by the group.
`Amr ibn Abasah reported that he heard the Prophet (peace and blessings be upon him) saying: “The closest that a slave comes to his Lord is during the middle of the latter portion of the night. If you can be among those who remember Allah, the Exalted One, at that time then do so.”
The number of rak`ahs to be performed during Tahajjud
Tahajjud prayer does not entail a specific number of rak`ahs which must be performed nor is there any maximum limit which has to be performed. It would be fulfilled even if one just prayed one rak`ah of Witr after the `Isha’.
Samurah ibn Jundub said: “The Messenger of Allah ordered us to pray during the night, a little or a lot, and to make the last of the prayer the Witr prayer.” This is related by at-Tabarani and al-Bazzar. Anas quoted Allah’s Messenger (peace and blessings be upon him), as saying: “Prayer in my mosque is equal to ten thousand prayers [elsewhere]. And prayer in the Sacred Mosque is equivalent to one hundred thousand prayers [elsewhere]. And prayer in the battlefield is equivalent to one million prayers [elsewhere]. And what is more than all of that is two rak`ahs performed by a servant [of Allah] during the middle of the night.”. This is reported by Abu ash-Shaikh, Ibn Hibban in his work ath-Thawab, and al-Munzhiri, in his book at-Targhib watTarhib, is silent about it.”