I would say that there are multiple reasons why people who have been bullied become bullies themselves. For some of them, it's because they think if they can find somebody weaker than them, they can finally have some street cred. For some, I think it's because they don't know any better.
I went to school with a kid who bullied me, and yes, he had an unhappy family life, and the like, but nothing changed his behaviour. I remember, one day, he was caned for flicking bits of orange peel through a biro casing, but when he came back to class (he sat in front of me) he was telling the person beside me what happened, how the deputy principal said to them, "All right, line up, and hold out the hand you don't write with." And I thought, later on, he hasn't learnt a damn thing from this.
The previous year, we were on a school excursion and a prison in our city was visible from the Freeway. He was sitting behind me on the bus and pointed to the prison and told us, with what seemed like pride, that his brother had nearly been sent there for stealing cars.