Why do we find evidence against null hypothesis in hypothesis testing? Why do you make the claim that you want to prove ( or have a hunch to be true) to be alternative hypothesis? One might argue that it has to do with the way hypothesis testing is set up. That you find evidence against null hypothesis as based on the definition: 
A null hypothesis is a statistical hypothesis that is tested for possible rejection under the assumption that it is true.

So the subsequent question is: Why do we find evidence against null hypothesis? Why not against alternative?

Because we are making an assumption that the null hypothesis is true, and are trying to prove that assumption by computing the probability a random sample follows that assumption. (And also the alternate hypothesis is the exact opposite of the null hypothesis.)
