I want to test whether a correlation coefficient is greater than 0.5.
What should the null and alternative hypothesis be?
Should I do a two tailed test or one tailed test?
One thing which may be difficult to determine using sampling theory is $P(data|\rho>0.5)$. I don't know of a sampling theory based decision problem which evaluates this directly (that is not to say it is impossible, I may just be ignorant).
You usually have to take a "minimax" approach. That is, find the "most favorable" $\rho_1$ within the range $\rho>0.5$, and do a specific test of $H_0:\rho=0.5$ vs $H_1:\rho=\rho_1$. Usually the "most favourable" value is the MLE of $\rho$, which is the sample correlation, so you would test against $H_1:\rho=r$. I believe this is called a "severe test" because if you reject $H_1$ in favor of $H_0$, then you will also reject all the other possibilities in favor of $H_0$ (because $H_1$ was the "best" of the alternatives).
But note that this hypothesis test is all under the assumption that the model/structure is true. If the correlation were to enter the model in a different functional form, then the test may be different. It is very easy to think that the test is "absolute" because usually you don't explicitly specify a class of alternatives. But the class of alternatives is always there, whether you specify it implicitly or explicitly. So if you were being pedantic about it, you should really include the full model specification and any additional assumptions in both $H_0$ and $H_1$. This will help you understand exactly what the test has and has not achieved.