Why does Voldemort use this spell?

Voldemort had something to prove FOR himself and TO his followers. Voldemort had figured out that it was love that made the first use of the killing curse fail. But his followers had their doubts, and Voldemort realized this as well.

So Voldemort made it a point to kill Harry himself, and to do it publicly so that his followers will conclude that Harry is not "the chosen one". And it had to be the same curse that failed the first time so that he could prove that his powers were the best in history (and that Harry really was not immune to the curse). But he did not know what Dumbledore knew.

Dumbledore, being just as extraordinary in his intellect as he was as a wizard, knew Voldemort's intentions. He had painstakingly put all the puzzle pieces together. And he had a plan to defeat Voldemort if Harry did exactly the right moves. Dumbledore figured out that Voldemort would in fact do what he did, AND that it would not work on Harry because he possessed part of Voldemort's soul. Furthermore, once done, that part of Voldemort's soul would be destroyed while not actually hurting Harry. That is why Dumbledore instructed Harry to surrender himself.

Harry and his friends made the rest of it work.

Another point of view is that Voldemort thinks that Harry survived to the Avada Kedavra because of the protection given by his mother sacrifice. In the Goblet of fire, when Voldemort return, he says

It was love. You see, when dear, sweet Lily Potter gave her life for her only son. . . . . .she provided the ultimate protection. l could not touch him. It was old magic. Something l should have foreseen. But no matter, no matter. Things have changed. l can touch you. . . . . .now.

So I guess that he is sure that now, without that "Old magic", the Avada Kedavra will work well.

Voldemort used Avada Kedavra because it was the strongest unforgivable curse. No one can survive it unless very strong magic protects that person. In the first movie of the series, Harry survived the spell because his mother protected him with a very old spell where her death will protect Harry. And because of that spell, Voldemort couldn't touch Harry. But in Harry Potter and the Goblet of Fire movie, Voldemort was back with Harry's blood which broke his mother's spell, and Voldemort could touch him. So Voldemort could easily finish Harry with the strongest unforgivable curse as the protection was no more. So Voldemort thought to finish Harry with this curse once and forever. Moreover, Voldemort has an evil mind and mostly evil minds like to curse rather than do a destructive magic like confringo which is used by many good wizards. Harry Potter has done this magic more than once. So he liked to do a more evil magic which is generally used by the Death Eaters. So he used this unforgivable curse which can't be changed once it is done.