On a literal level, Jack uses the clay to blend in with his environment. As a hunter, he has to disguise himself from his prey so that he may get as close as possible can to launch a successful attack. Jack wants to get nearer to the pigs and then kill them. Furthermore, the painting of his face affirms his status as a hunter. It differentiates him and the other hunters from the rest of the group. In their macro-cosmic society, they have a specific role to fulfil and their being painted signifies this.
In addition, wearing a mask symbolically empowers Jack and his hunters. They can hide their true nature behind the paint. The masks they wear enable them to hide their fears and become aggressive and fearless. Making their true identity invisible means that they can ignore that which is taboo. The fact that they are 'savages' means that the rules of civilised society do not apply to them any more. It becomes easier for them to kill, hurt and maim.
More specifically, for Jack, this signifies his transition from being just a choirboy to being someone important. He becomes more of a man, performing, as it were, a 'manly' task, which is hunting. He becomes a provider, someone who takes care of his own - this is, in itself, a source of great pride and actualisation. Painting his face, therefore, symbolically indicates his move from boyhood into manhood.
In this regard, then, it is clear from Jack's actions once he is thus disguised, that he feels stronger and more able to break the rules. He becomes more aggressive and volatile. He criticises the rules and ignores the conch. He ignores requests for civility and the performance of other duties, such as looking after the fire. He is more assertive and seeks others to join him. It is in this mode that he is eventually able to break away from the group and form a separate one.
Furthermore, on a deeper level, this act indicates Jack's descent into bestiality. He becomes more connected to his innate lust for blood. The innocent choirboy turns into a barbarian once he has put on his disguise. The rules of courtesy and civility no longer apply to him. He is, after all, in the wild and has to make up his own rules as he goes along.
The tragedy of this transformation is that Jack is driven more by the most basic of man's desires - to selfishly and recklessly assert his dominance whilst destroying everything that is good in the process. This is most pertinently illustrated by the frenzied slaughter of the pigs, Simon's death, Piggy's execution, the hunt for Ralph and the near complete destruction of the island when the fire burns out of control.
The arrival of the naval officer restores order but the damage, tragically, cannot be undone.