Basic due diligence on a startup offer is asking for # of shares outstanding, last company valuation, strike price. Advanced due diligence is talking about things like extended exercise windows, secondary sales, and liquidation preferences.
Unfortunately, basic due diligence is rare enough that if you do ask a potential employer the latter kind of question, there is a risk of coming off as overly mercenary.
The way of talking to potential employers that I've seen work is to ask questions in increasing complexity, sharing your conclusions along the way, and signalling why you're asking these questions.
After you ask the basic due diligence questions, you can share the math you're doing on stock value various exit scenarios (a good base assumption is to assume an exit at the current valuation).
That typically lays good groundwork for having "advanced" due diligence conversations about an extended exercise window and shows you're serious. In contrast, if the company isn't willing to share valuation or total share numbers, this is a huge red flag as it prevents you from doing the basic math.
This is a tool I built giving engineers the questions they need to ask, in order to do that basic math on what their stock is worth: http://www.optionvalue.io/