This is a question that requires far more than a simple 10-minute answer, as due diligence is an extremely complex subject and only having a few "quick tips" would put you in a very vulnerable position.
Generally speaking though, the key areas that you would want to focus on (depending on the type of the web business that you're about to buy) are:
* Financial verification - make sure to verify all income and expenditure, and never rely on screenshots or video proofs, as these are easily faked. Always require either live access to accounts/books or schedule a real-time screen sharing session with the seller.
* Make sure to fully understand the business model and its sustainability. This is easier said than done but it's perhaps the most important aspect of DD. You need to be able to make sure that the business is an actual, viable and sustainable business, rather than a fly-by-night website. I've written about this in length here: http://bryanoneil.com/the-most-important-website-due-diligence-question-that-buyers-rarely-ask/
* Take a very thorough look through the site's analytics (preferably you should request live access to its Google Analytics account) and make sure everything is in order. Also take a thorough look through the site's traffic sources and ensure that they're sustainable.
* Validate the claimed owner responsibilities so that you wouldn't end up buying a business that's actually a full time job. Sellers often misrepresent this part so it's important to perform a sanity check and ensure that the claimed hours match the reality.
But as I said, there's far more to due diligence than that. I've published a fair number of articles about this in my blog (http://bryanoneil.com) that you would probably find useful, but I would still recommend you to either do a lot of reading up on the subject, or to speak to a professional.
As for trustworthy brokers - I'm obviously a bit biased here as I run a brokerage myself (Deal Flow - http://dealflow.flippa.com/), but apart from us the other two larger brokers are Quiet Light Brokerage and FE International.
I'd recommend you to steer clear from brokers who either have too many listings (as that's an indication of sub-par vetting standards and therefore low quality listings), or brokers that haven't established themselves in the industry, as those newer brokers are rarely experienced enough to be able to properly validate the businesses that they list, and are often desperate to complete deals, leading them to intentional misrepresentation.
Hope this helps!