Since Voldemort was resurrected using Harry's blood, part of him would be Harry. Therefore, if he had succeeded with Avada Kadavra and killed Harry, would he have inadvertently killed the blood of Harry within him and therefore broken the very spell that allowed him to be alive? If not, would it have weakened him in any way?
Voldemort wasn't threatened in any way by using Avada Kedavra. Harry's blood helped give Voldemort a body but did not sustain him. The Resurrection Spell wasn't dependant on Harry being alive.
I think this is a fascinating and quite complex question. If I understand the question correctly, it's pondering whether Voldemort would have died had he succeeded in killing Harry, since he had used Harry's blood in his regeneration. My answer is no. If he had succeeded in killing Harry then Voldemort would have been unaffected by the fact that he took Harry's blood. The blood helped him regenerate; it did not sustain him. He needed it once and once only, not moment by moment. Harry's death would not have affected Voldemort's lifeforce since the spell which gave him a body had been performed many years beforehand.
Voldemort certainly tried to use Avada Kedavra on Harry on multiple occasions. The final confrontation between Voldemort and Harry at Hogwarts in Deathly Hallows (The Flaw in the Plan) referred to in the question is the only time they met where Voldemort actually had a chance of finishing Harry off (and vice versa, coincidentally). This is the only meeting between the pair where both characters are mortal. Before this moment, Voldemort was sustained by his Horcruxes and could not be killed. And Harry was sustained by the portion of soul embedded within him which tethered him to life whilst Voldemort lived.
"He took my blood," said Harry.
"Precisely!" said Dumbledore. "He took your blood and rebuilt his living body with it! Your blood in his veins, Harry, Lily's protection inside both of you! He tethered you to life while he lives!"
(Deathly Hallows, Chapter 35, King's Cross).
Note that Harry could not die as long as Voldemort lived. The same does not work in reverse. Voldemort did not depend on Harry's lifeforce in that way. Voldemort's immortality is dependent solely on his Horcruxes. He has no other hope of surviving death.
Of course, when he finally does use Avada Kedavra on Harry he comes down with a bad case of the jitters.
Voldemort seemed to be getting to his feet. Various Death Eaters were hurrying away from him, returning to the crowd lining the clearing. Bellatrix alone remained behind, kneeling beside Voldemort.
Harry closed his eyes and considered what he had seen. The Death Eaters had been huddled round Voldemort, who seemed to have fallen to the ground. Something had happened when he had hit Harry with the Killing Curse. Had Voldemort, too, collapsed? It seemed like it. And both of them had fallen briefly unconscious and both of them had now returned...
(Deathly Hallows, Chapter 36, The Flaw in the Plan).
So he falls unconscious whilst the portion of his soul within Harry dies. Then when Harry returns to consciousness Voldemort does too. This is the only damage Voldemort sustains. In all other regards he used Harry's blood as a one-time thing. Voldemort's body wasn't affected, whether Harry lived or died. All his blood does is give Voldemort a body.
"B-blood of the enemy...forcibly taken...you will...resurrect your foe."
(Goblet of Fire, Chapter 32, Flesh, Blood and Bone).
That's all the blood does. No more, no less.
It actually weakened him. You can remember from the movie that he falls down after avada kedavra, and avada kedavra actually breaks the spell. But he still has two horcruxes with him(the snake and himself) which die after the Avada Kedavra, so no he won't die and his weakness is as same as losing a horcrux.
Voldemort seemed to be getting to his feet. [...] Bellatrix alone remained behind, kneeling beside Voldemort.
Harry Potter and the Deathly Hallows - The Flaw in the Plan