Basis

What follows is how I apply the following guidance for administrators closing deletion discussions:

Wikipedia:Deletion policy: "These processes are not decided through a head count, so participants are encouraged to explain their opinion and refer to policy. The discussion lasts at least seven full days; afterwards, pages are deleted by an administrator if there is consensus to do so."
Wikipedia:Deletion process: "Consensus is formed through the careful consideration, dissection and eventual synthesis of each side's arguments, and should not be calculated solely by the balance of votes. Outcomes should reflect the rough consensus reached in the deletion discussion and community consensus on a wider scale. (While consensus can change, consensus among a limited group of editors, at one place and time, cannot override community consensus on a wider scale.)"
Wikipedia:Deletion guidelines for administrators: "Administrators must use their best judgment, attempting to be as impartial as is possible for a fallible human, to determine when rough consensus has been reached. (...) Consensus is not determined by counting heads, but by looking at strength of argument, and underlying policy (if any). Arguments that contradict policy, are based on opinion rather than fact, or are logically fallacious, are frequently discounted. For instance, if someone finds the entire page to be a copyright violation, a page is always deleted. If an argument for deletion is that the page lacks sources, but an editor adds the missing references, said argument is no longer relevant. (...) Wikipedia policy requires that articles and information comply with core content policies (verifiability, no original research or synthesis, neutral point of view, copyright, and biographies of living persons) as applicable. These policies are not negotiable, and cannot be superseded by any other guidelines or by editors' consensus. A closing admin must determine whether an article violates these content policies. Where it is very unlikely that an article on the topic can exist without breaching policy, policy must be respected above individual opinions. Per "ignore all rules", a local consensus can suspend a guideline in a particular case where suspension is in the encyclopedia's best interests, but this should be no more common in deletion than in any other area."

Step 1: Discard irrelevant opinions

I first identify opinions that are not helpful for determining whether there is consensus about the relevant issues. The relevant issues are whether and how to apply Wikipedia's policies, guidelines, past precedents or usual practices to the question of whether to delete, keep, or otherwise proceed with the article - or whether to make an exception in the interest of the project's goals). Opinions that are unhelpful for that purpose include:

Opinions that make no argument but are just a "vote" to keep or delete, or are limited to "per User:Foo", which amounts to the same thing. AfD is a discussion, not a vote, and all participants are expected to advance arguments of their own.
Opinions that plainly contradict core policies (e.g., "delete because it says bad things about our glorious leader" contradicts WP:NPOV).
Opinions that contain personal attacks, other misconduct, or allegations of such. AfD is not a dispute resolution forum and arguments about misconduct distract from the discussion's purpose.
Opinions that do not address the core arguments that have been advanced for or against deletion at all. For instance, in a debate about notability, this might include a "keep, it's useful" opinion or a "delete, not notable" opinion if other editors have cited several reviews in major newspapers.
Opinions that are too difficult to read, understand or find. For instance, I'm not going to search for the one relevant argument in a two-page mostly offtopic wall of text. I normally expect the essential arguments to follow a person's bolded "keep", "delete" or other recommendation.

Such opinions are not further taken into account.

Step 2: Count heads

Then I count how many remaining opinions there are for each proposed outcome such as delete, keep, redirect or merge. This provides me with a first approximation of the outcome of the discussion.

Step 3: Weigh arguments

Next I identify the principal lines of argument advanced for each outcome, and classify them roughly as follows:

Compelling: An argument that a core content policy mandates a certain outcome. For instance, per WP:BLP contentious material about living persons may not be kept unless reliably sourced, and per WP:C content must be released under a free license. If I conclude that one side has a compelling argument, this argument decides the outcome of the discussion.
Strong: An argument that is based on a straightforward and well-established application of a relevant policy or guideline - for example, deletion of an article about a historical person because no reliable sources exist to support their existence, or keeping an article about a politician who is a member of a national legislature. Such arguments carry particular weight (although it is possible that there are strong arguments for different outcomes).
Valid: An argument that is based on an interpretation of a policy, guideline or precedent that is at least defensible, and/or whose application calls for the exercise of editorial judgment. This applies mostly to questions about which experienced editors often disagree in good faith, such as whether sources are numerous, thorough or reliable enough for verifiability or notability. Because an AfD discussion (like other content discussions) is essentially the collective application of editorial judgment in curating Wikipedia, I as closer may not cast a "supervote" to influence this process; accordingly, I do not give such arguments either more or less weight.
Weak: This notably includes arguments generally agreed to be weak, insofar as they have not been already discarded, or possibly arguments that are otherwise particularly tenuous. Such arguments are given substantially less weight.

Step 4: Assess consensus

Then I determine whether rough consensus exists for any proposed outcome, taking into account the strength of arguments. As a rough guideline, if all arguments (that I have not discarded altogether) are about equal in weight, I will normally accept a majority of more than two-thirds as establishing rough community consensus for that particular outcome. But if one line of argument is particularly strong and the opposed arguments are particularly weak, I may consider that the strong argument represents community consensus even if it is put forth by as few as one-third of the discussion's participants, or fewer. This is because the prevailing argument is considered strong because it is based on a straightforward application of Wikipedia's policies and guidelines, which themselves are founded on a much broader community consensus than can be expected to emerge from any one AfD discussion.

Step 5: Attempt to reconcile outcomes

If no rough consensus is apparent after step 4, I check whether any of the proposed solutions have something in common for which there is consensus. For example, in a discussion about notability that is split roughly half in favor of deletion and half in favor of merging, both sides agree at least that there should not be a separate article about that topic. This I can implement with a closure to "redirect" the article to the proposed merge target. In this way, the normal editing process can then determine whether any content (and how much) should be merged from the redirected article's history.

If this process results in a rough consensus to do something, then I close the discussion accordingly. If not, the article is kept by default.