When asking "What's up with the title?", we're really asking two questions wrapped into one. We need to think about the novel's title, but there is also the series title to consider. So get ready; we're going to have to pull double-duty for this one:
Insane in the Crowned Membrane
Early in the novel, Tyrion Lannister comments, "All sorts of people are calling themselves kings these days" (3.Sansa.109). Those people specifically are Joffrey Baratheon, Robb Stark, Stannis Baratheon, Renly Baratheon, and later on Balon Greyjoy. This might not be a problem except these blokes are all claiming to be the lords over part of, or all of, the Seven Kingdoms, and that's not exactly the best way to get buddy-buddy with fellow sovereigns. (Or so we assume; the only thing we've ever lorded over is the local Pac-Man cabinet.)
As these five kings openly oppose one another, their battles and political maneuverings become known in the Seven Kingdoms as the War of the Five Kings. This war provides the major driving force for the novel's story, themes, and character conflicts, so its title, A Clash of Kings, naturally points toward it as being super important.
We should also point out that it is a clash not the clash of kings. In the series's first book, A Game of Thrones, the title suggested that the story was a singular political game but not the game of thrones. In other words, there would be more games to play regardless of who won or lost.
Similarly, this is simply one clash between kings. The history of Westeros is filled with clashes between monarchs, and it seems unlikely that this clash will put an end to the conflicts of the land. It's simply another rotation in a cycle of war and peace that has been revolving since the birth of the Seven Kingdoms.
Quick aside before moving on: Yes, Daenerys counts herself as queen of the Seven Kingdoms and Mance Rayder calls himself King-Beyond-the-Wall, but we don't count their stories as part of the clash referred to in the title. Their particular conflicts and stories are just too remote at this point and have little bearing on the struggle for the Iron Throne. Sure, that's what sequels are for, but those sequels won't be titled A Clash of Kings either.
A Long, Long Song
In A Clash of Kings, the series title, A Song of Ice and Fire, actually becomes part of the story itself. When Dany enters the House of the Undying, she sees a vision of a man, a woman, and their newborn child. The woman asks the man to write a song for the boy, and the man replies, "He is the prince that was promised, and his is the song of ice and fire" (49.Daenerys.40). Ladies and gentlemen, we have a title drop.
Dany later identifies the man as her dead brother Prince Rhaegar, and Jorah Mormont points out that he had a boy named Aegon, which suggests the woman was Elia Martell of Dorne. Jorah claims that Aegon was killed during the Sack of King's Landing, making any promises of his song being sung unlikely. Plus, Jorah has never heard of any song of ice and fire to boot, and he's a pretty learned dude (64.Daenerys.51-56).
By the novel's end, the song of ice and fire remains wrapped in mystery inside of a riddle stuffed in an enigma. It could be a prophecy, a song, or both, or nothing. Ha.
While its in-story purpose might be revealed in a later novel, we can say for certain that its thematic purpose as a series title is already pretty apparent by now. Consider what George R.R. Martin said about the series title during an interview:
People say I was influenced by Robert Frost's poem, and of course I was, I mean... Fire is love, fire is passion, fire is sexual ardor and all of these things. Ice is betrayal, ice is revenge, ice is… you know, that kind of cold inhumanity and all that stuff is being played out in the books. (Source)
We see all this being played out in the books. In the literal sense, the story contains fire-breathing dragons and the flame-loving Lord of Light countered by the icy Others and the coming of winter. In a more metaphorical sense, Martin's world is full of cold-calculating political maneuvers and fiery bouts of impassioned bloodshed—or as they refer to it in the Seven Kingdoms, the daily grind.
Also, let's pay attention to the fact that it's a song of ice and fire, not a song of ice versus fire. The "and" suggests to us that while these two opposites oppose one another—as opposites tend to do—they also come packaged together and are inseparable. Think of it as a yin-yang type of situation.
Fire and ice both exist together in the world and in the characters inhabiting it. For example, Jon's passion for duty could be his fire, but he keeps confronting the coldness of a reality that doesn't play by the rules, or that shuns him for his illegitimate birth. Tyrion Lannister is a man of serious sexual passions and love for his family, but the betrayals of his sister, father, and others in the court require him to coldly calculate against them. Finally, Catelyn finds herself constantly torn between her fiery love for her family and the cold necessity of her social duty.
As such, finding a balance between their personal fires and ices will be the conflict facing these characters, and may ultimately lead to their successes and failures alike. But we'll have to wait for later books in the series to say more on that.