Again, from 1 Samuel 16:1-3 (NASB):
Now the Lord said to Samuel, “How long will you grieve over Saul, since I have rejected him from being king over Israel? Fill your horn with oil and go; I will send you to Jesse the Bethlehemite, for I have selected a king for Myself among his sons.” 2 But Samuel said, “How can I go? When Saul hears of it, he will kill me.” And the Lord said, “Take a heifer with you and say, ‘I have come to sacrifice to the Lord.’ 3 You shall invite Jesse to the sacrifice, and I will show you what you shall do; and you shall anoint for Me the one whom I designate to you.”
My previous question was "would Saul actually have killed Samuel?" There's a fair amount of textual evidence that suggests that he would have, given his later violent and erratic behavior (combined with the fact that God gave Samuel a cover story rather than contradicting his prediction).
With that said, why did Samuel think that Saul would kill him? He was apparently correct to do so, but the violent and erratic behavior that the text records happened later. Still, the text doesn't directly mention any kind of threat against Samuel.
Verse 14 says "Now the Spirit of the Lord had departed from Saul, and an evil spirit from the Lord tormented him." (NIV) This verse is after Samuel had anointed David, but it's written in past tense (meaning that this may have happened before Samuel went to see David). Is this possible? If so, would Samuel have known about it, given that he wasn't seeing him at the time? Would this explain why he thought that this was a possibility?