x_0+x_1+\alpha x_2
\end{array}\right)$$ for a generic value of $\alpha$. For $\alpha\neq 0$ the entries form a regular sequence, since they are linearly independent linear forms. Therefore they define a complete intersection, that is a standard determinantal scheme, and $C$ is good determinantal.

Let $H$ be a general linear form. In particular we can assume that the coefficient of $x_3$ in the equation of $H$ is non-zero, so that $H$ does not contain the vertex of the cone $C$. Intersecting $C$ with $H$ we obtain a subscheme $X$ of ${\mathbb{P}}^3$, whose saturated homogeneous ideal $I_X$ is generated over $k[x_0,x_1,x_2,x_4]$ by the maximal minors of $$\left(\begin{array}{cccc}
x_0 & x_1+x_4 & 0 & x_2 \\
0 & x_1 & x_2 & x_0+x_1
\end{array}\right).$$ One can show that $X$ is good determinantal following the same steps as for $C$. Indeed, $C$ is just a cone over $X$.

Let $H=x_4$. Intersecting $C$ with $H$ we obtain a subscheme $Z$ of ${\mathbb{P}}^3$, whose saturated homogeneous ideal $I_Z$ is generated over $k[x_0,\ldots,x_3]$ by the maximal minors of $$\left(\begin{array}{cccc}
x_0 & x_1 & 0 & x_2 \\
0 & x_1 & x_2 & x_0+x_1
\end{array}\right).$$ $I_Z=I_P^2$ for $P=[0:0:0:1]$, hence $Z$ is a zero-dimensional scheme supported on the point 