The deaths (spoilers)

The deaths were symbolic of how he wronged people in his life.

Classmate - he bullied him, so he's tortured
Prostitute - he treated her like a piece of meat, so she was hung up with throat slit
Snitch - he beat him, so he was flogged to death with a scourge
Parents - they never saw him (because he didn't visit), so their eyes are gouged out
Partner - he betrayed him (or was going to), so he is literally stabbed in the back
Wife and daughter - he left them out in the cold with his whoring, so they froze to death
Himself as a child - cut apart bit by bit with his mistreatment of others

Of course, the case file that psychiatrist-priest/Pinhead had was his own.

In reality, he never killed anyone. Just pettily abused others like many people do. He opened the box in the bathroom and entered hell, never leaving. The whole movie takes places in hell - the dead classmate (and his own child-self's finger) was before the prostitute and his daughter never acknowledges his being there.

Much better than the campy third one. The cenobites have a much smaller role, like in the first film. (As has been pointed out, the script was changed to make it a Hellraiser film.)