I think you get carried away by the powerful formalism of projective geometry.
Actually the polynomial $f(x,y,z)$ does not correspond to a function $\mathbb P^2_k\to k$ , so that you can't define its differential. This is because, if the degree of $f$ is $d$, we have for all $\lambda\in k$ the equality $f(\lambda x,\lambda y,\lambda z)=\lambda^d f(x,y,z)$, and since $[x:y:z]=[\lambda x:\lambda y:\lambda z]\in \mathbb P^2_k$ there is no reasonable way to define $f(x,y,z)\in k$.
However the polynomials $\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y},\frac{\partial f}{\partial z}$ have three perfectly well defined zero sets, curves whose common points indeed are the singularities of $C(f)$.
But all is not lost. To $f$ you can associate an affine cone $V=V(f)\subset \mathbb A^3_k$ whose equation is $f(x,y,z)=0$. The polynomial $f$ defines a perfectly regular function $f:V\to k$ i.e. $f\in \mathcal O (\mathbb A^3_k)$. Then for all $P\in \mathbb A^3_k$ we have indeed $df_P=\left(\frac{\partial f}{\partial x}(P), \frac{\partial f}{\partial y}(P), \frac{\partial f}{\partial z}(P)\right):k^3=T_P(V)\to k$, and for $P\in V$, the vanishing of $df_P$ tells you whether $P$ is singular on $V$.
For example if $degree(f) \gt 1$, the origin is a singular point of the cone $V$, as it should be.