Here's a basic summary of the backstory you're looking for, Chknptpie. Spoilers (some of this stuff hasn't been detailed explicitly yet in the TV series, yet) but nothing major. Safe for non-book readers who want to know more of the backstory:
The Targaryen family ruled the Seven Kingdoms for about 300 years. The last king was Aerys II, called the "Mad King" because of occasional bouts of madness. His eldest son, Rhaegar, absconds with Lyanna Stark, Eddard's older sister and Robert Baratheon's betrothed.
At this point, Brandon Stark, Eddard's oldest brother and betrothed to Catelyn Tully, goes to King's Landing to confront Rhaegar. Aerys has him and Eddard's father, Rickard, killed, and demands that Jon Arryn turn over his wards Eddard and Robert. Arryn refuses; he and Robert call his banners. Eddard marries Catelyn in his brother's place, and musters theirs and his forces.
A bunch of battles are fought between the rebels, led by Robert, and the various Royalist forces that notably don't include those of Tywin Lannister, then the Hand of the King. At the Battle of the Trident, Robert kills Rhaegar and the Royalist armies more or less dissolve. Tywin takes King's Landing through deceit and Jaime kills Aerys.
The Queen, with Viserys, had been sent to Dragonstone. There, she gives birth to Daenerys and dies. Before the island is captured, Viserys and Daenerys escape across the Narrow Sea and begin their exile. Robert marries Cersei Lannister after Lyanna dies, and is crowned King. So basically Viserys feels that he is the rightful King, and that Robert has usurped him and driven him from his home.