Mukhtasar Bukhari

Umm al-Mu'mineen Aisha Siddiqa (may Allah be pleased with her) says that she never saw the Messenger of Allah (peace and blessings be upon him) performing the Tahajjud prayer sitting down, until when the Prophet (peace and blessings be upon him) became older, he would recite (Qur'an) while sitting, and then when he intended to bow (for ruku'), he would stand up and recite about thirty or forty verses and then perform ruku'.

Umm al-Mu'mineen Aisha Siddiqa (may Allah be pleased with her) says in a narration that he used to do the same in the second rak'ah as well. Then when the Prophet (peace be upon him) completed the prayer, he would see: if I was awake, he would talk with me, and if I was asleep, the Prophet (peace be upon him) would remain lying down.