The original "key verses" of the Sermon on the Mount, on which the entire sermon hangs but especially chapter 5, are 5:17-20:
"Do not think I have come to destroy the Law and the Prophets. I have not come to destroy them but fulfill them... unless your righteousness exceeds that of the scribes and Pharisees, you will never enter the kingdom of heaven."
The rest of chapter 5 plays this general statement out in its particulars. "You have heard... but I say to you." In other words, Jesus fufills the Law and the Prophets by giving the full story on things they have heard. Chapter 6 plays out what the sermon might mean by a righteousness that exceeds that of the scribes and Pharisees.
Meanwhile, the Beatitutes and loose exhortations to be salt and light set the stage, set the feel for the sermon as a whole, introducing the emotional tone of the sermon, although not so much the ideological stage.