"The south" designates Egypt, as it does in Daniel 8:9. "The king of the south," therefore, is the king of Egypt. Ptolemy I was "one of Alexander the Great’s favorite generals.... After Alexander’s death in 323 B.C., he seized Egypt as his share of the divided empire. He assumed the title of king of Egypt" (World Book Encyclopedia). And so Ptolemy became "the family name of the last dynasty of ... ancient Egypt. They ruled from 323 to 30 B.C." (Ibid.)
"The north" designates Babylon (Jeremiah 25:9; Ezekiel 26:7; Jeremiah 1:14; 4:6; 6:1, 22; 10:22; Daniel 8:4), so "the king of the north" is the one that rules over Babylon. During the time of the Greek empire, that would have been the Seleucid kings. The World Book Encyclopedia describes the Seleucid Dynasty as "A line of kings descended from Seleucus, a general of Alexander the Great. The dynasty began when Seleucus seized Babylon in 311 B.C."
"He that cometh against him" in verse 16 is Rome.
"The king" introduced in verse 36 is the papacy.