Step 1: Understanding primary valence.
Primary valence, according to Werner's coordination theory, corresponds to the oxidation state of the central metal atom. It is satisfied by negative ions and is ionisable in nature.
Step 2: Oxidation state of cobalt in CoCl$_3$.
In cobalt(III) chloride, cobalt is bonded to three chloride ions, each having a charge of $-1$. To balance the total negative charge, cobalt must have an oxidation state of $+3$.
Step 3: Relation to primary valence.
Since the oxidation state of cobalt in CoCl$_3$ is $+3$, the primary valence of cobalt is also 3.
Step 4: Conclusion.
Hence, the correct answer is (C) 3.