The three meanings are very similar, and are often used synonymously.
To assume something (to make an assumption) is to take something for granted. To believe something because you feel it is obvious or self-evidently true, rather than because you have evidence.
To suppose something (to make a supposition) can mean the same as to assume. However, is has more of a sense that you are aware the supposition might not be true. A supposition is more of a tentative or provisional belief than an assumption.
A hypothesis can mean the same as a supposition - a conditional belief. However it is usually used specifically to mean either (1) something taken to be true for the sake of argument, or (2) a proposed explanation for a scientific or academic problem or phenomenon.