How to Seek Laylat-ul-Qadr

Laylat ul-Qadr is the most blessed night. A person who misses it has indeed missed a great amount of good. If a believing person is zealous to obey his Lord and increase the good deeds in his record, he should strive to encounter this night and to pass it in worship and obedience. If this is facilitated for him, all of his previous sins will be forgiven.
Praying Qiyaam

It is recommended to make a long Qiyaam prayer during the nights on which Laylat ul-Qadr could fall. This is indicated in many hadeeths, such as the following:
Abu Tharr (radhiallahu `anhu) relates:
"We fasted with Allah's Messenger (sallallahu `alayhi wa sallam) in Ramadaan. He did not lead us (in qiyaam) at all until there were seven (nights of Ramadaan) left. Then he stood with us (that night - in prayer) until one third of the night had passed. He did not pray with us on the sixth. On the fifth night, he prayed with us until half of the night had passed. So we said, 'Allah's Messenger! Wouldn't you pray with us the whole night?' He replied:
'Whoever stands in prayer with the imaam until he (the imaam) concludes the prayer, it is recorded for him that he prayed the whole night.'…" [Recorded by Ibn Abi Shaybah, Abu Dawud, at-Tirmithi (who authenticated it), an-Nasa'i, Ibn Majah, at-Tahawi (in Sharhu Ma`an il-Athar, Ibn Nasr, al-Faryabi, and al-Bayhaqi. Their isnad is authentic.]
[Point of benefit: Abu Dawud mentioned: "I heard Ahmad being asked, 'Do you like for a man to pray with the people or by himself during Ramadan?' He replied, 'Pray with the people.' I also heard him say, 'I would prefer for one to pray (qiyaam) with the imaam and to pray witr with him as well, for the Prophet (sallallahu `alayhi wa sallam) said: "When a man prays with the imaam until he concludes, it is recorded that he prayed the rest of that night." [Masaa'il]]
Abu Hurayrah (radhiallahu `anhu) narrated that the Messenger (sallallahu `alayhi wa sallam) said:
"Whoever stands (in qiyaam) in Laylat ul-Qadr [and it is facilitated for him] out of faith and expectation (of Allah's reward), will have all of his previous sins forgiven." [Al-Bukhari and Muslim; the addition "and it is facilitated for him" is recorded by Ahmad from the report of `Ubaadah Bin as-Samit; it means that he is permitted to be among the sincere worshippers during that blessed night.]

Making Supplications

It is also recommended to make extensive supplication on this night. `A'ishah (radhiallahu `anha) reported that she asked Allah's Messenger (sallallahu `alayhi wa sallam), "O Messenger of Allah! If I knew which night is Laylat ul-Qadr, what should I say during it?" And he instructed her to say:
"Allahumma innaka `afuwwun tuh.ibbul `afwa fa`fu `annee - O Allah! You are forgiving, and you love forgiveness. So forgive me." [Recorded by Ahmad, Ibn Majah, and at-Tirmithi. Verified to be authentic by Al-Albani]

Abandoning Worldly Pleasures for the Sake of Worship
It is further recommended to spend more time in worship during the nights on which Laylat ul-Qadr is likely to be. This calls for abandoning many worldly pleasures in order to secure the time and thoughts solely for worshipping Allah.
`A'ishah (radhiallahu `anha) reported:
"When the (last) ten started, the Prophet (sallallahu `alayhi wa sallam) would tighten his izaar (i.e. he stayed away from his wives in order to have more time for worship), spend the whole night awake (in prayer), and wake up his family." [Al-Bukhari and Muslim]
And she said:
"Allah's Messenger (sallallahu `alayhi wa sallam) used to exert more (in worship) on the last ten than on other nights." [Muslim]