1 Cersei knows Robert loved Eddard's sister
Cersei knows that King Robert , her husband, had only one true love which is Eddard Stark's sister.
2 Arya stark attacked Joffrey
Arya stark's wolf injured Joffrey then Arya disarmed him and threw away his sword. Joffrey hated Arya and then his mother fueled that hatred. See Summary of Episode 2 season 1.
3 Bran knows about Jaime and Cersei
Jaime Lannister wanted to kill bran Stark because he saw him with Cersei. See Summary of Episode 1 - Season 1.
4 King Joffrey thinks Eddard is a traitor
King Joffrey was convinced by Cersei that Eddard Stark is a traitor. See Summary of Episode 9 - Season 1.
5 Eddard Stark told Stannis the truth
Eddard Stark sent a raven to Stannis telling him that he is the true Heir for King Robert. Cersei and King Joffrey hated Eddard more for that. See why Eddard Stark died.
6 Robb stark held Jaime Lannister captive
Robb stark held Jaime Lannister captive during the war. This led Tywin Lannister to hate the starks.
7 The Starks went for war against the Lannisters
Robb stark raised an army and started attacking the Lannisters. He won many battles and so became more hated by the Lannisters. See Summary of Episode 10 - season 1.
8 The Starks held on to ethics
Eddard stark held on to ethics in many situations and this went against the plans of the Lannisters who act based on their wants without being bound by anything.
9 Jaime hates Eddard
Jaime hates Eddard as he judged him when he killed the mad King. See why Jaime killed the mad king.