The Five Kings are:
King Joffrey ... ahem... Baratheon: He thinks the Iron Throne is his by rights, being Robert's eldest son. Of course, we know that he is NOT Robert's son! He is backed by the Lannisters, which are the wealthiest family in the kingdoms. They are trying to get an alliance with Dorne (one of the kingdoms that has yet to make an appearance) by marrying Princess Marcella to the Prince of Dorne's youngest son.
King Stannis Baratheon: Robert's eldest brother. He has the best "legal" claim to the throne since Cersei's three children are bastards. So, he's also claiming the throne "by rights". He is angry with Renly for trying to step over him in the succession and he does not want to ally with Robb Stark, since Robb wants to take away half of his kingdom.
King Renly Baratheon: The youngest of the Baratheon brothers. He's claiming to be king because he thinks he's better for the job than his older brother, Stannis. Renly points out that Robert didn't have any real "legal" claim to the Iron Throne, he took by war. He's rejecting the succession out of hand. Catelyn is going to parley with him about an alliance. Right now, he's got the biggest army.
King Robb Stark: The King in the North. Robb really doesn't want to sit on the Iron Throne. He's fighting for independence for the North. It hasn't been real clear in the show, but he is claiming the Riverlands as part of his kingdom as well. Success for him would be for whoever wins the Iron Throne to recognize the North/Riverlands as a separate kingdom.
King Balon Greyjoy: The "king" of the Iron Islands wants to be king again. He's going to try to take the North, as he described to his children. Remember, he tried to rebel against King Robert at one point, but lost the war, so he had to give his son, Theon, to the Starks as a ward/hostage. He's not really after the Iron Throne, but he wants to take a big bite out of the North.
Hope that helps!