Success is what actually turns Saul against David, not failure. The women's song after the victory — "Saul hath slain his thousands, and David his ten thousands" (1 Samuel 18:7) — is the spark, and from that day Saul watches David "with an evil eye." What follows is close to a decade of David as a hunted man, and it's worth noticing what he does and doesn't do with the opportunities he's given. Twice — once in a cave at En-gedi, once in Saul's own sleeping camp — David has a clean, undetectable chance to kill the man trying to kill him, and twice he refuses, on the same principle both times: "the LORD forbid that I should do this thing unto my master, the LORD's anointed" (1 Samuel 24:6; 26:9–11). Notice what he's actually protecting there. It isn't Saul's life particularly — it's the principle that a throne taken by murder is a throne with no legitimate foundation under it, and David seems to understand, years before he'll need the lesson himself, that how you get power matters as much as whether you get it.
This is also where you meet Jonathan, Saul's own son and heir, who should by every political logic have been David's rival and instead becomes his closest friend — making a covenant with him, stripping off his own robe and armor to give to David, warning him repeatedly of his father's plots at real personal cost (1 Samuel 18:1–4; 20). Jonathan's loyalty to David over his own inheritance is one of the quieter, more remarkable relationships in the Old Testament, and it costs Jonathan his life alongside his father's at Gilboa — a loss David will mourn with a lament (2 Samuel 1:19–27) as raw as anything else he ever wrote.