A nony mouse: There are various places in Tanakh in which the existence of life after death is alluded to or mentioned. Look at Tehillim 17, for example, or 1 Samuel 28.
Belief in gigulim appears in the Zohar, the Ramban, the tzfat Kabbalists, and virtually every major rabbi to discuss the subject in the past 500 years (with the exception of the Rashash). Yet because some pre-publication-of-the-Zohar rishonim were against the concept of gilgulim, one is not obligated to believe in it. Some also say that one is not obligated to believe in any particular kabbalistic concept; it is all seen as optional.
But reincarnation is not just a Hindu belief — it is found in various religions throughout the world.