The sample space $\Omega=\{(x,y)| x,y\in\{1,2,3,4,5,6\}\}$. So, $Supp[X]=Supp[Y]=\{1,2,3,4,5,6\}$.
Let $Z=|X-Y|$. Naturally, $Supp[Z]=\{0,1,2,3,4,5\}$. The distribution of $Z$ is computed in the following table:
| $z\in Supp[Z]$ | the event $(Z=z)$ | $P(Z=z)$ |
|---|---|---|
| 0 | $\{(6,6),(5,5),(4,4),(3,3),(2,2),(1,1)\}$ | $\frac{6}{36}$ |
| 1 | $\{(5,6),(4,5),(3,4),(2,3),(1,2),(2,1),(6,5),(5,4),(4,3),(3,2)\}$ | $\frac{10}{36}$ |
| 2 | $\{(4,6),(3,5),(2,4),(1,3),(3,1),(4,2),(5,3),(6,4)\}$ | $\frac{8}{36}$ |
| 3 | $\{(3,6),(2,5),(1,4),(4,1),(5,2),(6,3)\}$ | $\frac{6}{36}$ |
| 4 | $\{(2,6),(1,5),(5,1),(6,2)\}$ | $\frac{4}{36} $ |
| 5 | $\{(1,6),(6,1)\}$ | $\frac{2}{36}$ |