In this chapter, David mourns Saul's death.
This chapter begins the story again without so much as a preamble. We pick up with David's reaction to the death of Saul, and like so many other things in the OT, David commemorates Saul's life with a song. (For other songs, consider the song of Deborah, the song of Moses, another song of Moses, and so on.)
Let us analyze the Amalekite's story. The Amalekite says that he came upon Saul when Saul was near death. This is interesting because 1 Samuel 31 (the chapter that we just read before this one) tells us that Saul "fell on his own sword" and died, and then his armorbearer did the same thing and died. It would appear that the Amalekite is not telling the truth, or at least he is not telling us the same thing as the author of Samuel. So that raises the question, how did the author of Samuel discover the ultimate fate of Saul?
1 Samuel 31:7 gives us a hint; it says that "the men of Israel ... saw that the men of Israel had fled and that Saul and his sons were dead." So it's plausible that some of the Israelite survivors of the battle saw how Saul had died, impaled on his own sword, still wounded by Philistine arrows, and must have inferred that he was injured and killed himself to avoid capture.
The second question raised is this: is the Amalekite lying, or is his story somehow complementary to what the author of Samuel wrote in the previous chapter? It is conceivable that Saul "fell on his sword" and then survived somehow only to request an attending Amalekite to slay him. However, 1 Samuel 31:5 makes it pretty clear that the armorbearer slew himself when he saw that Saul was dead. So all these things taken together, I think it's unlikely that the Amalekite is telling the truth.
This raises the third question, which is this: why would the Amalekite lie, when David kills him for it? And the answer to this question is more clear than the previous. The Amalekite, like so many others, assumed that David was a mortal enemy of Saul and would reward him for killing the Hebrew king. This was a common perception at the time, and we see Achish showing similar reasoning. Achish assumed that David would fight on his behalf against Israel. This was a perception that David encouraged when he told Achish that he was taking his men to attack Israelite and Kenite towns.
Basically, the Amalekite thought he was bringing "good news" to David, and that David would be pleased with him. Obviously, this was not the case, and the Amalekite loses his life for it. Verse 16 puts it clearly: David was really, really loyal to "the LORD's anointed". Even when the "LORD's anointed" was a psychopath who was trying to kill David and led Israel into several disastrous defeats.
The second half of this chapter is David's song. This is called "the song of the bow" (v. 18), and even more interestingly it is "written in the book of Jashar". The book of Jashar (or Jasher) is an ancient Hebrew book that has no extant copies. Which is to say, nobody alive in the last 1000 years has ever read it, and probably longer than that. It is possible that the book of Jashar is also referenced in Joshua 10:13. I always find these kinds of books fascinating, where we can read about them but cannot read the book itself because it does not exist anymore. We can guess based on its description here that it might be a book of poetry or war songs or something. But really, this one reference is basically everything we know about the book of Jashar.
David's song itself is basically a poetic retelling of what we just read at the end of 1st Samuel. David obviously loved both Saul and Jonathan, and the "mountains of Gilboa" that David is cursing is where Saul and Jonathan died. I don't really have anything else meaningful to say about this part, it's just another poetic song.