Analysis of Argument
Paragraph 1
I always start begin with a broad statement about the issue at hand. For instance in an argument essay about whether it was necessary to conserve energy, I would begin with:

Energy conservation has become a hot topic in the United States as of late. There are those who debate climate change and those who affirm it, those who assert that humans are depleting the world's natural resources, including fossil fuels, and those who believe that humans are barely making a dent in consuming the resources the earth has to offer.

This sets up a fluid introduction to my essay, and by starting with an introductory sentence, I felt I would set myself apart from the majority of other test takers who begin "I agree" or "I disagree" -- remember, you want to distinguish yourself from other essay writers (in a good way) right from the start.

I always continue by saying, "In the preceding statement, the author claims that (paraphrase of the author's argument)." This shows that I understand the author's argument. I continue with my disagreement (and, as has been often said, ALWAYS disagree with the author's reasoning, and I pretty much always used the same thesis statement, something like:

Though his claim may well have merit, the author presents a poorly reasoned argument, based on several questionable premises and assumptions, and based solely on the evidence the author offers, we cannot accept his argument as valid.

I can use that statement for pretty much any argument essay I encounter. The beginning shows that I see the issue as nuanced, rather than black and white, and I recognize that though the author may have arrived at the correct answer, his reasoning is flawed. I also use "we" but I try to avoid using "I". Take that as you will.

Paragraph 2
For me, paragraph two always attacks the premises of the author's argument. I usually jot down a couple of premises on my board -- these are things that show up in the author's argument, usually without any evidence provided. So 99% of the time, I attack the lack of evidence that the author bases his premises on. I usually begin with some version of:

The primary issue with the author's reasoning lies in his unsubstantiated premises.

I like "primary" and "secondary" as transitional tools because they are more complex than "first" and "second". Throughout the first paragraph, I show flaws in the author's premises by pointing out his lack of evidentiary support (they pretty much always lack evidentiary support) and where they are open to holes or alternate explanations. I usually have two or three, but one would be fine if it it was strong. I generally end with something like:

The author's premises, the basis for his argument, lack any legitimate evidentiary support and render his conclusion unacceptable.

Paragraph 3
In the third paragraph, I always attack the assumptions - again, I jot down a couple assumptions on my notepad while reading the prompt. Generally, the issue with assumptions is that they need to be clearly explicated - the author is asking the reader to make a jump with them, but the reader may well veer off course if the author doesn't explicitly state his arguments. I usually begin with some variation of:

In addition, the author makes several assumptions that remain unproven.

Again, I begin with a transitional phrase that the e-grader can pick up on. As with the premises, I spend this paragraph attacking a couple of the author's assumptions. The easiest way to do this is to find an alternate explanation -- ie, what if the assumption wasn't true? I usually have two or three, again. My assumption paragraph ends with something like:

The author weakens his argument by making assumptions and failing to provide explication of the links between X and Y he assumes exists.

Paragraph 4
Paragraph 4 is where I talk about how the author could strengthen his argument -- that is, I go back to my claim that his argument could have some validity, but not as it stands. I usually begin this paragraph with something like:

While the author does have several key issues in his argument's premises and assumptions, that is not to say that the entire argument is without base.

Then I provide some concrete ways the author could strengthen his argument. The easiest way to do this is to give examples of what kind of evidence the author could provide, and discuss how he can fill the holes in his assumptions. I generally end with something like:

Though there are several issues with the author's reasoning at present, with research and clarification, he could improve his argument significantly.

Paragraph 5
This is my conclusion paragraph. I pretty much always conclude with the same sentence:

In sum, the author's illogical argument is based on unsupported premises and unsubstantiated assumptions that render his conclusion invalid.

I usually use "in sum" because it's considered better stylistically than "in conclusion" but signals to the e-grader that you're at your conclusion. I usually add a couple sentences of fluff in between and then I end with:

If the author truly hopes to change his readers' minds on the issue, he would have to largely restructure his argument, fix the flaws in his logic, clearly explicate his assumptions, and provide evidentiary support. Without these things, his poorly reasoned argument will likely convince few people.

...And that's pretty much it.

So the cliff notes:

P1- Intro with generic thesis statement that works for 99.9% of argument essays.

P2 - Attack the premises of the argument.

P3 - Attack the assumptions of the argument.

P4 - Discuss what type of evidence or reasoning would strengthen the argument.

P5 - Conclusion.

Hope that helps someone out there and good luck on your essays!