Show that $k[x,y,z]/(xz-y^2)$ is not a UFD.

It's easier just to observe that $xz=y^2$ in your ring gives a decomposition into irreducibles that is not unique ($x$, $y$, and $z$ are irreducible because they are of degree one; your quotient is homogeneous). Of course, this also shows $x$ is not prime since it divides $y^2=y y$ but divides neither $y$ nor $y$.

The algebra $k[x,y,z]/(xz-y^2)$ is a graded domain, with (the classes of) $x$, $y$ and $z$ all in degree $1$.

In a graded domain, the product of two non-homogeneous elements is non-homogeneous. So if $x$ is a product of two elements, these must be homogeneous. Since $x$ has degree $1$, it can only be a product of an element of degree $1$ and an element of degree $0$. Since the only elements of degree zero are units, we are done.