Step 1: Draw the bonding around carbon.
In phosgene, $\text{COCl}_2$, the central carbon forms a double bond to oxygen and one single bond to each of the two chlorine atoms. Carbon thus uses all four of its bonds and keeps no lone pairs.
Step 2: Recall the formal charge formula.
Formal charge $\text{FC} = V - N - \dfrac{B}{2}$, where $V$ is the free-atom valence electrons, $N$ is non-bonding (lone-pair) electrons, and $B$ is the number of shared (bonding) electrons.
Step 3: Find $V$ for carbon.
Carbon is in Group 14, so $V = 4$.
Step 4: Find $N$ for carbon.
Carbon has no lone pairs in this molecule, so $N = 0$.
Step 5: Find $B$ for carbon.
Carbon makes 4 bonds total (a double bond counts as 2 bonds plus 2 single bonds = 4 bonds). Each bond is 2 shared electrons, so $B = 4 \times 2 = 8$.
Step 6: Compute the formal charge.
$\text{FC} = 4 - 0 - \dfrac{8}{2} = 4 - 4 = 0$. The carbon carries zero formal charge, which is option (4).
\[ \boxed{\text{Formal charge on C} = 0} \]