Throughout the last two chapters, Paul talks about how we have a pretty big problem -- sin (AKA anything that counters what God wants, which almost always ends up screwing over other people and even ourselves). In fact, after addressing some counterpoints, Paul concludes that "both Jews and Greeks [...] are all under sin" (3:9).
Then Paul quotes some of the Psalms that further talks about how all people have their own sin problem.
Paul talks about how there is "none righteous" (3:10). The verse is from Ecclesiastes 7:20, which talked about how no one "does good and does not sin" (and the NIV translates "does not" as "never"). While many of us have done good deeds before, some of our deeds are not good, and that's basically the point. Righteousness by God's standards is doing good (or at least not doing bad) all the time, and unfortunately, we can't really accomplish that by ourselves.
Paul also talks about how there is "none who understands" and "none who seeks after God" (3:11). The verse is either from Psalm 14 or Psalm 53, which is both about David finding issue with the evil in the world. Note that from the original psalms, the claim was made by God to see if anyone would seek him, so it's implied that these two instances are regarding people by themselves. There are multiple instances of people seeking after God, but they do so because of the drawing of God, Jesus, and/or Holy Spirit (and Jesus talks about how he draws mankind to himself in John 12:32 -- he may be doing this so we can do what we can't do by ourselves!).
Paul then talks about how mankind "become[s] unprofitable" and how there is "none who does good" (3:12). These two verses are also from Psalm 14 or Psalm 53. There are some questions since I'm pretty sure we can think of a time someone "did good", but it's worth distinguishing man's standards from God's standards and noting that the context of the psalms implies an overall state of being (given that God looks widespread). God's standard of "doing good" in the context of overall state of being is basically following God's standards all the time with a pure heart -- something that none of us can do. It's kinda like how some students say they didn't do good on a test, even though they got some questions right (though God's passing score is basically 100%). Also, according to The Free Dictionary, "do good" can also mean "be beneficial for", and us having a sin problem isn't very beneficial for God who does not like any sin (since it ruins lives). So even if we could do good in one particular instance, we failed to do good in life overall by God's standards (and many of us can't even match man's standards!).