The first step to understanding this paragraph is to understand the context. At this point in the story the person who Harry cares about the most has just been killed, and Dumbledore is in the process of taking the blame for it.
In the preceding paragraphs Dumbledore explained to Harry how not giving Harry all the information earlier indirectly led to this tragedy. At this point Dumbledore attempts to soften the self-criticism. This paragraph is a potential justification for what he did. Now let's take it one sentence at a time:
Is there a defence?
I don't think there are any language issues with this sentence. It is simply introducing what is to come, namely the possible defence for Dumbledore's actions.
I defy anyone who has watched you as I have — and I have watched you more closely than you can have imagined — not to want to save you more pain than you had already suffered.
This sentence is made more complex by the parenthetical remark between the dashes, so let's remove that for now leaving us with:
I defy anyone who has watched you as I have not to want to save you more pain than you had already suffered.
We can further simplify the fundamental structure of the sentence by removing the phrase "as I have" since that is only modifying the immediately preceding phrase. So now we have:
I defy anyone who has watched you not to want to save you more pain than you had already suffered.
If there is a particular word here that is giving you trouble, it is likely "defy". Here is the Merriam-Webster definition (3):
to challenge to do something considered impossible : DARE
So in essence Dumbledore is stating that it would be impossible for anyone who has watched Harry not to want to save him pain. I.e. anyone who watched Harry would want to save Harry pain. This is part of Dumbledore's justification for his actions — he did what anyone would have done.
Now let's add back the parts we took out. The phrase "as I have" is a slight limitation to the claim just formulated. It's not quite anyone who watched Harry that would fell/act this way; it's only those who have watched Harry in the same way that Dumbledore has watched Harry. However, that does not give us enough information, since we still don't know what the definition of "in the same way that Dumbledore has watched Harry" is.
For this we have to add the parenthetical between the dashes. This phrase gives us the definition we need:
and I have watched you more closely than you can have imagined
So now we understand that the limitation is to only those people who have watched Harry as Dumbledore did, which is to say people that have watched Harry more closely than Harry can have imagined. So essentially Dumbledore is saying that anyone who has watched Harry more closely than Harry could imagine, would want to spare Harry more pain. Note, that in the context of the story this is theoretical. There aren't actually any other people who have watched Harry in this fashion; Dumbledore is saying that any theoretical person who would have watched Harry in this fashion would have felt the same way.
Now the next sentence:
What did I care if numbers of nameless and faceless people and creatures were slaughtered in the vague future, if in the here and now you were alive, and well, and happy?
This sentence is somewhat tricky. First, let's discuss the purpose of the sentence. this sentence is continuing to explain the motivation for Dumbledore's actions, but it presents a justification and a condemnation at the same time. The justification is that it allowed Harry to be alive, well, and happy. The condemnation is that it risked the lives of many other people. This is a sort of moral dilemma, and Dumbledore is showing both sides of the issue. He wants Harry to understand that even though what he (Dumbledore) did was probably wrong, there is at least something of a case that can be made for it.
Now let's look at the actual words and structure of the sentence. There are actually two different grammatical ways to parse this sentence. To understand this, let's remove everything from the second "if" and on which leaves us with:
What did I care if numbers of nameless and faceless people and creatures were slaughtered in the vague future
This is a rhetorical question. Dumbledore is explaining that his actions were consistent with not caring about the lives of other people. The word "if" here represents a condition, or more precisely a possibility. Dumbledore is saying that the result of his actions might have been the future slaughtering of others, but he went ahead with his actions anyway. Notice, that he makes it seem not as bad by using the terms "nameless", "faceless", and "vague future". One can be more easily excused for causing the slaughter of theoretical people at a theoretical time than for causing the slaughter of real people right now. Thus, this is part of the justification.
When we reattach the other half of the sentence it can be one of two things. It can either be a separate condition/possibility, or it can be a condition/possibility within the condition/possibility already mentioned. To illustrate the first way, we can remove the first "if" clause, to leave us with:
What did I care if in the here and now you were alive, and well, and happy?
This would be an additional condition/possibility that Dumbledore would be expressing his lack of care for. I.e. Dumbledore would be saying that there were two conditions/possibilities that he did not care about. He did not care if other people would get slaughtered, and he did not care if Harry would be alive, well and happy.
From the context of the discussion we can reject this reading, because we know that Dumbledore's whole point is that he loved Harry too much. As such, he would not be saying that he did not care if Harry would be alive, well, and happy.
This brings us to the other way of reading the sentence, where the second "if" is a condition/possibility within the first "if". In this case Dumbledore is not making an absolute statement that he didn't care that others would be slaughtered. He is saying that he didn't care because of a specific condition/possibility, i.e. Harry being alive, well, and happy. What he is actually saying is that he's okay with Possibility A because within Possibility A there is Possibility B, but if not for Possibility B than he would not be okay with Possibility A.
In other words, this sentence is showing us that Dumbledore had to weigh two competing concerns. On the one hand, his actions might cause other people to die, but on the other hand they might cause Harry to be alive, well, and happy. Dumbledore is saying that he was willing to risk the former precisely because of the possibility of the latter. So the two "if"s are telling us that if Dumbledore would do something that might cause A it might also cause B, and that's why he was willing to do it.
This presents the justification and the condemnation simultaneously — he justifies his actions by appealing to how they helped Harry, but at the same time he condemns them for how they harmed others.
Now for the last sentence:
I never dreamed that I would have such a person on my hands.
This sentence is a little vague because it doesn't specify who/what "such a person" is. If we try to use the previous sentence to figure it out, the only person discussed there is Harry and he is simply described as "alive, well, and happy". This sentence then doesn't seem to be so necessary for Dumbledore's point. However, what he probably means is that he had never even dreamed that Harry would be alive, well, and happy, and therefore when he had the opportunity to enable it he was more easily swayed because it was something that he had previously thought was so out of reach.
So in sum, the paragraph as a whole consists of Dumbledore's defence of his actions, partly interwoven with an implied condemnation of his actions. The double "if" structure of the third sentence refers to a possibility within a possibility.