How was Harry going to kill Sirius Black?

In "The Prisoner of Azkaban", chapter 17, "Cat, Rat, and Dog":

Harry raised the wand. Now was the moment to do it. Now was the moment to avenge his mother and father. He was going to kill Black. He had to kill Black. This was his chance...

How was Harry going to kill him, had he tried to go through with it? It's only his third year, and I don't think he knows any killing curses yet.

Answer

Canonically speaking, I'm going to say that Harry wasn't going to kill Sirius at all. Yes, Harry was blind with anger and hurt, and he dreamed of avenging his parents by killing Sirius, but Harry had not been taught any deadly curses prior to Goblet of Fire and he didn't understand the emotional component of killing another human being. As Bellatrix Lestrange said to Harry, after he failed to cast the Cruciatus curse against Bella in Order of the Phoenix, "You've got to really mean it, Potter!" [Executing the Unforgivables, that is/paraphrase]

However, Harry's experience with any kind of magical killing curse prior to Goblet of Fire is limited to the following:

‘Then kill him, fool, and be done!’ screeched Voldemort.

Quirrell raised his hand to perform a deadly curse, but Harry by instinct, reached up and grabbed Quirrell’s face –

‘AAAARGH!’

_{Philosopher's Stone - pages 213-214 - Bloomsbury - chapter seventeen, The Man With Two Faces}

The "deadly curse" isn't even named! Prior to Goblet of Fire, Harry had not heard of Avada Kedavra.

How Harry knew Quirrell was going to perform a deadly curse -- since Quirrell only raised his wand without saying anything -- is not explained. In canon, aside from Avada Kedavra, no other curse or spell is described as being deliberately lethal, except, I'd argue, Fiendfyre (And even then it's a stretch; it is lethal magic, but its foremost purpose is to burn and destroy; that it is lethal is incidental.).

Before all the talk about Harry wanting to kill Sirius in Prisoner of Azkaban, Harry is determined to hurt Sirius, to try and offload the hurt Harry himself felt due to his parents' death. Harry wanted Sirius to hurt, emotionally, as badly as he did.

He had forgotten about magic – he had forgotten that he was short and skinny and thirteen, whereas Black was a tall, fullgrown man. All Harry knew was that he wanted to hurt Black as badly as he could and that he didn’t care how much he got hurt in return ...

_{Prisoner of Azkaban - page 249 - Bloomsbury - chapter seventeen, Cat, Rat, and Dog}

Harry was not trained in magic well enough to kill Sirius. If he were to actually try (Which I feel is against his characterization. Anger? Yes. Murder? No. Killing is something deep down that Harry would never want to do, although I don't think he regretted killing Voldemort.) I think he would have tried to physically kill Sirius. After all, the thought of magic had fled Harry's mind and he and Sirius had an initial physical altercation. Yes, Harry then pointed his wand at Sirius, but if magic had fled his mind, raising his wand could have merely been reflexive.

Anyhow, there are basically only two ways Harry could have killed Sirius and canon only allows for generalizations: Harry would have killed Sirius either by magic or through physical violence. Order of the Phoenix demonstrates Harry isn't capable of an Unforgivable at age 15, much less 13. There are no canon instances showing Harry physically beating anyone to the point of death. So it's a toss up.

_{Personally, I don't think Harry would have ever killed Sirius, especially at age thirteen. I do think canon ultimately supports this.}