The Hindu tradition of reincarnation predates the Buddha, and he was brought up in the belief system. It is a different belief system though in that, as reflected in caste, each person has theri station in life. If they do their duty well, then they will be reincarnated into a higher level. This is a very rigid system, still prevalent in India and causes no end of problems. It limits what little social mobility there is, and means that if you are born a rag picker, a rag picker you are likely to stay.
The Buddha's system of reincarnation is about personal karma and the motivation - good and bad for acts committed in life. Whereas the Hindu system has the belief in a soul - Atman - The Budha's does not. Of course o-one can prove the veracity of this teaching, but the Advanced practitioners report that this is so, and describe an intermediate state in Tibetan Buddhism called the Bardo, which is a bit like a dream world between lives. There are practices which can take the practitioner back through theri previous lives.