Interesting question. The good news is that the Hebrew of King David’s time is actually closer to modern Hebrew than that of the Second Temple period (with its considerable Aramaic influences). This is partly due to the deliberate efforts of the Zionist leadership to hark back to the nation’s heroic past, and partly because, in the revival of Hebrew in the modern era, the narratives of the Hebrew Bible provided far more source material than the Second Temple period, when the Talmudic Sages (who were virtually the only ones putting things down in writing) tended to slip into Aramaic all the time.
As a result, the glimpses of dialogue that we see in David’s time sound remarkably contemporary. Two examples, out of many: