Indeed a very good question
First, the time of tahajud ends when fajr times starts or few minutes before
And you there is no special way of offering the prayer however you should make the niyyah for tahjud and make sure you make dua after praying tahajud because that's the time when duas are accepted because its has been reported by Abn Hurairah
The Messenger of Allāh, peace and blessings of Allāh be on him, said,
"Our Lord, blessed and exalted is He, descends every night to the nearest heaven when the latter one-third of the night remains, (and) says, Is there any one who calls upon Me so that I may accept of him, who asks of Me so that I may grant him, who seeks forgiveness of Me so that I may forgive him?
I hope this answers you question.