It is not clear that Jimmy Porter himself knows what he wants. Osborne himself observes in his description of Jimmy that to be this vehement is "to be almost non-committal." In other words, if you attack a specific aspect of society, it is clear what you want, but if you attack everything, as Jimmy does, this seems like a general, directionless expression of ill-temper.
Jimmy's anger is often misdirected, particularly at Alison, but his constant railing at her, and her background in particular, does provide some evidence for his particular source of dissatisfaction and therefore what he wants. Jimmy is driven largely by class resentment. He believes that he is more intelligent than, and just as well-educated as, the upper classes who run the country, but he has no power and nothing to occupy him. What he wants, therefore, is a disruption in the social order to allow him and (rather less importantly) those like him to occupy positions of power and influence and to find an outlet for their talents.
This would be a political solution for the sense of inferiority and inadequacy he feels. On a more individual, psychological level, he probably also wishes to stop feeling like such a strange hybrid: poor and working class but also sensitive and highly-educated. However, he is less willing to acknowledge this desire and is only partly aware of it himself.