I just found an interesting account in 1 Samuel that I don't remember ever noticing before. If I understand the story line correctly, it is telling us that Uriah wasn't the first time David committed murder (or at least killed) to cover up another sin.
In 1 Samuel 27:8-12, David has gone to live with the enemy, the Philistines in Gath with Achish. While there, David was given Ziklag as a town and place to live along with his followers. According to 1 Sam 27:8, David made raids against the various Canaanite tribes in the region and lied to Achish about where he had made raids. When Achish questioned David, David lied and told him the raids had been in Judah or against other parts of Israel (1 Sam 27:10). Then verse 11 gets pretty graphic: "David would leave neither man nor woman alive to bring news to Gath..." (1 Sam 27:11).
This raises a lot of questions about God's purpose and how much leeway David had on his way to becoming king of Israel. I've got some study and some thinking to do to figure it all out, especially how this points to Christ.
But it is clear that the events surrounding Bathsheba were not the first time David had used violence to cover up his actions.