(I'm not sure the term *exact monoidal category* is a standard one in the literature, so I'll just assume I know what you mean by it.)

Yes, $K_n$ commutes with products, as some have mentioned, but the tensor monoidal functor $\otimes : \mathcal C \times \mathcal C \to \mathcal C$ is not exact, so it doesn't induce a map $ K_n \mathcal C \times K_n \mathcal C \to K_n \mathcal C$. Rather, the functor $\otimes : \mathcal C \times \mathcal C \to \mathcal C$ is *bi-exact*, in the sense that it is an exact functor in each of its two variables, separately. (That's what makes $K_0 \mathcal C$ into a ring.)

The result is that the collection of abelian groups $ K_n \mathcal C$, indexed by $n$, forms a graded ring, with products $K_m \mathcal C \otimes K_n \mathcal C \to K_{m+n} \mathcal C$. Assuming the tensor product is commutative up to natural isomorphism, the ring is commutative in the graded sense that $ x y = (-1)^{mn} y x$.

The actual construction is intricate, the main idea being to construct a map $K \mathcal C \wedge K \mathcal C \to K \mathcal C$ of spectra. See, for example, Waldhausen's approach in section 9 of Algebraic K-theory of generalized free products. I, II. Ann. of Math. (2) 108 (1978), no. 1, 135–204. It uses the Q-construction of Quillen, but could be simplified by using the S-construction of Segal that Waldhausen developed extensively.

My paper with Gillet, The loop space of the Q-construction, Illinois Journal of Mathematics, 31 (1987) 574-597, gives another approach that avoids both spectra (i.e., delooping) and the phony multiplication trap that Steve mentioned.

PS: My paper Algebraic K-theory via binary complexes, Journal of the American Mathematical Society, 25 (2012) 1149-1167, gives an algebraic description of the elements of the higher K-groups that converts the construction of the product into a simple exercise involving tensor product of chain complexes.