You answered the question yourself: MOST champions fight for a city-state. Some don't. I think that probably about 50% of the champions in the League are only there for personal goals. Clearly, it's alright to join the League for yourself . To answer your specific champion questions:
Alistar uses his status as a champion to speak out against Noxus and those they tread upon. I would assume that he was one of the more vocal advocates of the Noxus-Ionia rematch.
Blitzcrank is only in the League because it was the only place where he could exist without being nothing more than a curiosity.
Gragas is in the League because a few Summoners saw his immense combat prowess and invited him to join. He agreed, though I don't know how drunk he was beforehand to have agreed to that
Lulu joined for a similar reason as Blitzcrank. She doesn't represent any one cause, though I would assume she would defend Bandle City if necessary.
Malphite fights in the League as a way of showing his support for its mission and to keep the peace between the city states. Same with Nasus.
Orianna is a robotic clone of Corin Reveck's daughter, who desperately wanted to be a champion of the League but who was killed during her training. He built the robot Orianna as a way for his daughter to live out her dream.
Rammus is here for the same reasons as Lulu and Blitzcrank.
Renekton IS being held captive and participates in the League because the summoners have nothing else to do with him. I think he rather enjoys it though, since he can kill to his heart's content.
Ryze participates in the League in order to study powerful magical creatures and mages that work there.
You're absolutely right about Shaco. No one knows why he's here
Veigar represents himself. He seeks world domination, and he figures the easiest way to begin his bid for power is at the locus of Runeterra's politics - the League of Legends.
Vladimir was recruited by Noxus due to his immense hemomantic powers.
Xerath is there for himself. He wishes to use the magical energies of the nexuses to finish breaking the magical sarcophagus that binds him, but he can't get close without drawing the ire of the Summoners. So, he pretends to fight for them while seeking a way to tap into the nexuses that they guard.
No one knows why Jax is here. He was recommended as a candidate by Reginald Ashiram, but since Ashiram disappeared shortly after making that comment, no one knows just why he was recommended.