It is a conjunction, introducing a subordinate clause. From Wiktionary:
Conjunction: 1. Introducing a clause that is the subject or object of a verb (such as one involving reported speech), or that is a complement to a previous statement.
He told me that the book is a good read.
By analogy, "It is a proof that the book is a good read."
Generally, if you can replace the word "that" with "which" without changing the meaning, then "that" is acting as a pronoun; also, if you can replace "that" with "that thing" or "that event", then it's also a pronoun. In this case, you would get:
* If I see an article advertised, it is the surest proof I know which the article does what is claimed for it.
* If I see an article advertised, it is the surest proof I know which the article does that thing is claimed for it.
As an aside, do you intend the word "article" to mean "thing", or to mean "length of text written about something"? If the former, it would be much more natural to use the word "something" or "an object". "Article" in English is only really used to mean "length of text" or more rarely "item in a list".