I interpret your question to mean the following: there is a functor, which I'll write $G \mapsto LG$, sending a group $G$ to its set of conjugacy classes. Is it possible to lift this functor to a group-valued functor?
The answer is no. To prove this, it suffices to find an inclusion $H \to G$ of groups inducing an inclusion $LH \to LG$ of conjugacy classes, but such that the order of $LH$ doesn't divide the order of $LG$ (so it can't be a group homomorphism with respect to any group structure on $LH$ or $LG$). A very small example suffices: the inclusion $C_2 \to S_3$ induces an inclusion $LC_2 \to LS_3$, but the former has order $2$ and the latter has order $3$.
(However, $LG$ is naturally a groupoid-valued functor; namely, it should really be thought of as returning the action groupoid $G/G$ of $G$ acting on itself by conjugation. The geometric realization of this thing is, among other things, the free loop space of $BG$.)