Tikkun chatzot is recited almost by definition in the middle of the night. I know far more people who are still awake at that time (generally somewhere between 12 AM & 1 AM) than who sleep before it and wake up for it. So why do texts that discuss saying tikkun chatzot talk about waking up for it, rather than staying awake until chatzot, reciting tikkun and then going to bed until shacharit?
EDIT: Shulchan Aruch Orach Chaim 1:2:
ב הַמַּשְׁכִּים לְהִתְחַנֵּן לִפְנֵי בּוֹרְאוֹ, יְכַוֵּן לַשָּׁעוֹת שֶׁמִּשְׁתַּנּוֹת הַמִּשְׁמָרוֹת שֶׁהֵן בִּשְׁלִישׁ הַלַּיְלָה וּלְסוֹף שְׁנֵי שְׁלִישֵׁי הַלַּיְלָה וּלְסוֹף הַלַּיְלָה, שֶׁהַתְּפִלָּה שֶׁיִּתְפַּלֵּל בְּאוֹתָן הַשָּׁעוֹת עַל הַחֻרְבָּן וְעַל הַגָּלוּת, רְצוּיָה.
One who rises to pray before his Creator should aim for the watches of seasonal hours, which are at 1/3 through the night, 2/3 through the night and the end of the night, because the prayer one prays at those times over destruction and exile are more accepted.