While David was rescuing his family from the Amalekites, another Amalekite was on the way to find David with news from the battlefield where Saul had died. This is a good time to review a mistake that Saul had made early on. God had commanded him to utterly destroy all the Amalakities... but, Saul had stopped short of that objective. Obviously, their population had been rejuvenated because David was encountering them at every turn. Now, the Amalekite who came to David with news from the warfront thought he had news that would cheer David up and cause David to reward him. Boy, oh boy! Was he ever wrong!
Obviously, the Amalekite did indeed stumble upon the body of Saul on the battlefield. He had Saul's crown and bracelet to prove it. But, it seems to me that his claim that he finished Saul off must be false. Saul had fallen on his own sword and died in front of his own armor-bearer (I Samuel 31:4-5). That lie cost the Amalekite his life.
So, David was faithful to his king all the way to the end of Saul's life. In a sense, David's first kingly act was to execute the man who claimed to have killed the former monarch. And, from II Samuel 1:17 on, we find the record of David's sincere sorrow over the loss of Saul and Jonathan. He lamented, wept and mourned because of the passing of these royal Israelites. He cursed the land upon which they died. He spoke highly of them and of their feats. Especially for Jonathan; David was brokenhearted because of his death. It is an especially great person who can mourn the passing of his enemy.