Did Saul anoint David as king of Israel?

No. Depending on the source you prefer, either Samuel anointed David, or the elders of Israel anointed him. In 1 Samuel, God was angry with King Saul for disobeying Samuel's instructions to utterly destroy the Amaleks and all their animals, so Samuel secretly anointed David to replace Saul as king. The Book of Chronicles contains no mention of Samuel's role in anointing Saul as king - the elders of Israel simply joined in anointing David king by acclamation, in gratitude for his leadership under Saul.