Why did isriallites Believe David was there greatest king?

This is because he fought golaiath and he was brave and trusted God and was a very good role model to them

As king, David was a warrior who protected the land of God's people, and Jehovah saved David wherever he went. During his reign, the nations boundaries were extended from the river of egypt to the river euphrates. David's reputation lived on well after his death, even today.