@Pablo R.
You're actually on the right track. You want them to open up to you. The key thing is to let talk and ask open ended questions, like:
Why are you moving?
Do you live in this neighborhood already?
Where do you live now?
How is your current landlord? <---all the complaints come out on this one.
Then describe your rental criteria and look for them to flinch: no evictions, 3X rent in income...
Then casually say something like "You know, we're just trying to get an accurate picture of who we are dealing with. We know it's hard out here and we don't want to take an application fee from someone and who doesn't meet the rental criteria. We want honesty above all else. Don't tell me you have a pristine rental history then I see 4 evictions and a felony on your background report. " Then wait and listen This is where they feel compelled to tell you their deep, dark secrets and disqualify themselves.
Yes, if they say something that disqualifies them in your book, then absolutely mark them off the list. Their honesty is what you want, but it does not obligate you to allow them to rent your unit.