1 Cersei knows Robert loved Eddard's sister
Cersei knew about her husband, the king 'Robert' that he was deeply in love with Eddard Stark's sister 'Lyanna.' Cersei envy Lyanna, which was one of the reasons that she hated Starks.
2 King Joffrey thinks Eddard is the traitor
Cersei was able to convince the king Joffrey that Eddard Stark is a traitor and can deceive them.
3 Arya Stark attacked Joffrey
Joffrey starting hating Arya with the incident when Arya's wolf injured Joffrey. While he was injured, Arya disarmed him and threw away his sword, which made Joffrey developed hatred. Joffrey's mother, Cersei, fueled this hatred and made all of the Lannisters against Starks.
4 Bran Stark knows about Jaime and Cersei
Bran was fond of climbing walls. He was doing the same and saw Cersei and Jaime together, making love. Jaime did not want everyone to know about their relationship and hence, wanted to kill Bran Stark. (See Why did Cersei get arrested?)
5 Eddard Stark told the truth
One of the reasons that Cersei and Joffrey felt animosity towards Starks was when Eddard sent the raven to Stannis telling him; he is the true Heir for King Robert.
6 Robb Stark held Jaime Captive
The fight between The Lannisters and The Starks took a new turn when Jaime was held as a captive by Robb Stark due to which Tywin Lannister started hating Starks. (See Why did Robb Stark die in Season 3?)
7 The Starks went for War against The Lannisters
Robb Stark was one of the greatest fighters as he won many battles. He initiated the attack against the Lannisters due to which they hated Robb intensely.
8 The Starks held on to ethics
Eddard Stark fought with ethics and did not forget them, no matter how hard the situation was. This habit of Eddard made Lannisters to fall apart from their plan.
9 Jaime felt animosity towards Eddard
Jaime hated Eddard as he was judged by Eddard when he killed the mad king.