How can I determine which of two complementary hypotheses should be the null? How can you determine the direction of the test by looking at a pair of hypotheses? How can you tell which direction (or no direction) to make the hypothesis by looking at the problem statement (research question)?
 A: Since we test assuming that the null is true, the null has to include a way to determine a sampling distribution.  This means that if one of the hypotheses includes an $=$ and the other does not, then the one with the $=$ is the null.  For example if the hypotheses are $\mu \ge 0$ and $\mu < 0$ then the 1st is the null and the 2nd is the alternative.
If there is not a clear $=$ then the null generally represents the ideas of "no difference", "no change", or "status quo" while the alternative is generally one of "there is a difference" or "things have changed".
The alternative is usually what we would like to show and the null is what we want to reject, so there are cases where the equality can be reversed (with some additional assumptions), for example we may want to show that a new cheaper method is just as effective as the old method, so we are trying to show equality.  For this we do an equivalence study (without infinite data we cannot prove equality, but we can show equivalence if we decide on a range of values we consider to be equivalent).  In this case the equal condition is part of the alternative that we are trying to show and the null is of a difference (but in practice we test with the null at the equivalence boundaries, so it still has equality, actually we usually just see if the CI is within the equivalence range).
Your question is very general and the answer can change with the details.  If you want a more precise answer then we need a more detailed question.
A: There is no clear way to tell which should be the null hypothesis, and which should be the alternative, but it can be a fateful decision.
Suppose we have two hypotheses


*

*$H_F$: Ferraris are faster than Lamborghinis

*$H_L$: Lamborghinis are faster than Ferraris


and we want to decide between them given a set of relevant data $D$.
If we make $H_F$ the null, then the hypothesis test consists in:

  
*
  
*Calculate the likelihood, which is the PDF over all the possible data given $H_F$: $p(d\,|H_F, I)$.
  
*Find the (say) 0.95-HDR (Highest Density Region) of the likelihood, and call it $R_F$.
  
*If $D \in R_F$, conclude Ferraris are still faster; else, conclude Lamborghinis are faster.
  

Imagine also a similar test with the roles swapped between the car brands.
Now, it is eminently possible that $D \in R_F$ ($\rightarrow$ keep $H_F$) and $D \in R_L$ ($\rightarrow$ keep $H_L$)! In other words, with the same data $D$, the conclusion can depend entirely on which hypothesis is designated the null. It is also eminently possible that $D \notin R_F$ ($\rightarrow$ reject $H_F$) and $D \notin R_L$ ($\rightarrow$ reject $H_L$), which are equally inconsistent.
More concerning still, the hypothesis test does not refer at all to the alternative hypothesis. So in the case above with $H_F$ as the null, if the alternative hypothesis were "Ferraris are slower than snails" and $D \notin R_F$, the test would direct us to reject $H_F$ in favour of the absurdity.
