Here's what a good modern commentary says (Word commentary series):
"But it is clear, especially from what follows in the sermon, that despite the language used this is not to be understood quantitatively (contra Luz)that is, that the righteousness Jesus speaks of does not come through a greater preoccupation with the minutiae of the law that outdoes even the Pharisees! The ethical teaching presented by Jesus in the Gospel can hardly be said to do that. Instead, Jesus expects, as the antitheses to follow show, a new and higher kind of righteousness that rests upon the presence of the eschatological kingdom he brings and that finds its definition and content in his definitive and authoritative exposition of the law. Thus Jesus clearly calls his disciples to a way of righteousness, but it is a new way that rests upon the true meaning of the Torah now delivered by the Messiah. To follow that teaching is to follow the path that leads to perfection (5:48)."
That is, Jesus is calling for a better kind of righteousness, not more of the same thing.
Calvin's comment is actually fairly similar.
"But it deserves inquiry, whether he does not rather blame the corrupted manner of teaching, which the Pharisees and Scribes followed in instructing the people. By confining the law of God to outward duties only, they trained their disciples, like apes, to hypocrisy.2 They lived, I readily admit, as ill as they taught, and even worse: and therefore, along with their corrupted doctrine, I willingly include their hypocritical parade of false righteousness. The principal charge brought by Christ against their doctrine may be easily learned from what follows in the discourse, where he removes from the law their false and wicked interpretations, and restores it to its purity. In short, the objection which, as we have already said, was unjustly brought against him by the Scribes, is powerfully thrown back on themselves."
What does this have to do with "Calvinism?" I'm not sure how directly it is connected. The ideas here are held widely by Protestants, not just Calvinists. This passage is an introduction to a set of interpretations by the 10 commandments. In them Jesus replaces external rule-keeping (which the scribes and Pharisees emphasized) with a focus on intent: caring about other people.
So this is really about the kind of lives people who are justified by faith live. You'll never enter the kingdom is you try to get there by following rules, "works of the Law." Rather, people in the Kingdom live lives based on loving God and our fellow man.
Note that "righteousness" isn't always a technical term involved with imputation. The core meaning in the OT and NT is living the way God wants, but depending upon context may also mean being right with God. In Romans Paul says that God considers us to right with him when we have faith (that he imputes faith as righteousness). But those who are justified by faith are still responsible for living according to Jesus' teachings, which is what righteousness is about here.