How to perform Tahajjud prayer

Praise be to Allah, the Lord of the World; and blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

There is no difference between Tahajjud and Qiyam al-lail (the late night prayer). If the prayer is performed after one had slept then it is called Tahajjud otherwise it is called Qiyam-al-lail.
There are many evidences in Qur'an and Sunnah proving the excellence and merits of Tahajjud.
Allah Says (interpretation of meaning): {Is one who is obedient to Allah, prostrating himself or standing (in prayer) during the hours of the night, fearing the Hereafter and hoping for the Mercy of his Lord (like one who disbelieves)? Say: "Are those who know equal to those who know not?" It is only men of understanding who will remember (i.e. get a lesson from Allah's Signs and Verses).}[39:9]. Allah Praises those who perform the late night prayers "Their sides forsake their beds, to invoke their Lord in fear and hope, and they spend (in charity in Allah's Cause) out of what We have bestowed on them. No person knows what is kept hidden for them of joy as a reward for what they used to do.}[32:16-17].
Imam Tirmizi narrates from Abdullah Ibn Salaam that "When the Prophet (Peace and blessing be upon him) came to Madinah "When the Prophet (Sallallahu Alehi Wasallam) came to Madinah, 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 are sleeping and you will enter paradise in peace."' This is related by al-Hakim, Ibn Majah, and at-Tirmidhi who calls it Hasan Sahih.
The Prophet said: "The best prayer after the obligatory prayers is the prayer which is performed in the last part of the night". [Reported by Imam Muslim]
Therefore, the night prayer is a confirmed Sunnah, its rule does not change even in the month of Ramadhan. However, performing it during Ramadhan is better than during other months.
The time of Tahajjud (Qiyam al-lail) starts after obligatory prayer of Isha and extends till dawn. There is no specific time for this prayer so it may be performed at any time of night but the preferred time is the last third of the night.
There is no specific number of Rakah of this night prayer but the Prophet never prayed more than thirteen Rakah during Ramadhan or any other time.
The Sunnah is to be performed in sets of two Rakah. It is recommended to shorten the first two Rakah making the remaining Rakah longer. And then one should end it with one Rakah of Witr. One may also make Dua Qunoot in Witr if he likes.
The person who does not memorize much Qur'an may recite many short Surahs in one Rakah or he may read looking in the Mus-haf too.
Allah knows best.