This is the format in which I was given the PMF.
Given this pmf
$$\begin{array}{lll} x&y&f_{xy}(x,y)\\ \hline 1&1&.25\\ 1&2&.25\\ 2&1&.25\\ 0&0&.25 \end{array}$$
How to find the Correlation coefficient? I thought about making the formula
$$\frac{1}{4}\text{ from intervals }0\leq x \leq 2, 0\leq y \leq 2$$
But it makes less and less sense as i move on since i don't even have all the values for the pmf. Any help?
The distribution is discrete. Since the given probabilities add up to $1$, the "missing" entries are all $0$, so are not missing at all.
To compute the correlation coefficient, you will need the covariance and the two variances. Let us start. From the table, $E(XY)=(1)(1)(0.25)+(1)(2)(0.25)+(2)(1)(0.25)+(0)(0)(0.25)$.
Now let us compute $E(X)$. This can be done directly from the table, but you might prefer to first find the pmf of $X$. From the table, we have $\Pr(X=1)=0.25+0.25=0.5$, $\Pr(X=2)=0.25$, and $\Pr(X=0)=0.25$. From this you can find that $E(X)=(1)(0.5)+(2)(0.25)+(0)(0.25)=1$. Calculation of $E(X^2)$ is also straightforward.
When we look at $Y$, we get a pleasant surprise. Now it is a matter of putting the pieces together.