Albert Barnes Commentary 2 Samuel 1:10

"So I stood beside him, and slew him, because I was sure that he could not live after that he was fallen: and I took the crown that was upon his head, and the bracelet that was on his arm, and have brought them hither unto my lord." — 2 Samuel 1:10 (ASV)

The Amalekite was one of those who came to strip the slain on the morrow after the battle (1 Samuel 31:8), and was lucky enough to find Saul and take possession of his crown and bracelet. He probably set out immediately to seek David and invented this story, possibly after hearing an account of what really happened from an Israelite prisoner.