Tahajjud - you can make it anytime after the witr. Best time is last third of the night. You just need to make intention for Tahajjud and after that it's the exact same procedure as any other nafl salat.
Qaza - you make intention for qaza of a certain salat, then make it like the fardh portion of that salaat. Qaza is only required for fardh and witr salats.