What exactly was Mao trying to accomplish during the Cultural Revolution?

There were two main objectives of the cultural revolution.

The first of which was for Mao to put himself back into the center of power. After the failure of the great leap forward, Mao had being sidelined from power by his designated successor Liu Shaoqi and Deng Xiaoping. Both of whom valued pragmatic, Soviet style planned economy over Mao's more radical economic policies. As such, the Chinese economy underwent a series of stabilization in the mid-1960s in reaction to the failed policies of the GLF. Mao however, wanted to once more to be in the driver's seat of China and hence the cultural revolution started out as a ploy to purge his political opponents at the center of Chinese politics and re-assert himself as the supreme leader. And indeed Deng and Liu were purged very early on.

The second of which was a genuine desire to transform Chinese society and to train a new generation of revolutionary leadership who could be relied upon to continue Mao's revolutionary project. Had the cultural revolution only being about a political struggle at the top of the leadership structure, then it would have ended in 1966 after Mao had purged his opponents in the Politburo.

However Mao believed that the entire party structure had gotten complacent and corrupt, and that it would sooner or later need to be replaced by a new generation of revolutionary leaders. The way to "train" the new revolutionaries is to, of course, have them "make their own revolution" even if it is against the state and party he had built. It was also directed against the stagnation and corruption he perceived in both the party and society. The violence and chaos of the cultural revolution was, in other words, a way for the masses to carry out a second revolution on the road towards true Communism.

In an ironic way Mao was the last great Chinese conservative. He was probably the last person who wished to find a "unique" Chinese path towards modernization before Deng decided that the better path would be to follow the capitalism of the rest of the world.