I understand the historical background of the "five points" (as responses to the Remonstrants), however, I think Calvinism would do much better starting where Calvin did, with the knowledge of God and the knowledge of man. Too much is lost by just starting at "grace".
I don't see how starting with duplex knowledge tells people what Calvinism is. How does that summarize the content of Calvinism? Now, the 5-point rubric is definitely limited. It highlights what is distinctive to Calvinism in contrast to classical Arminianism. It basically summarizes the canons of Dort. There is, of course, much more to Calvinism. A more representative sampling is the Westminster Shorter Catechism, not to mention the Westminster Confession of Faith. However, the uninitiated might find the quaint language of the WSC a barrier to comprehension, while they might be lost in the WCF–which takes for granted a basic grasp of historical theology and traditional terminology.
I'm not saying the five points aren't helpful. But they provide distinctives that have roots farther up (earlier in) systematic theology. Sovereignty is one of them. What "grace" is depends on what "sovereignty" is, for example.