How do I determine if this argument is valid?
The argument $\frac{P}{C}$ means "If we assume that $P$ is true, we can prove that $C$ is true." For example, $\frac{A}{A\lor B}$ because, if you can prove that $A$ is true, then $A\lor B$ is also true: there's no way that $A$ can be true but $A\lor B$ can be false at the same time.

In the argument you've been give, $P$ and&nbsp;$C$ are completely unrelated. There is a way that $P$ can be true and $C$ can be false at the same time: specifically, $P=\mathrm{true}$, $C=\mathrm{false}$. Therefore, the argument is not valid.

You don't calculate the conclusion, per se. rather, you look for a situation in which the premise is true but the conclusion is false. If you can find such a pair of premise and conclusion, the argument is invalid. Conversely, if you can prove that the premise being true always makes the conclusion true, the argument is valid.

I don't understand what your teacher is trying to say. It's not enough to show that the argument is valid for one particular instantiation of&nbsp;$P$ and one particular instantiation of&nbsp;$C$. "Valid" means it works for all instantiations. Also, the examples of $P$ being "I am thirsty" and $Q$ being "I drink some water" doesn't work.  There are situations where you are thirsty but you don't drink water (maybe you drink nothing; maybe you drink juice), and situations where you're not thirsty but you do drink water (maybe you're Mark Zuckerberg at a Senate hearing), as well as situations where you're thirsty and drink, or not thirsty and don't drink.