No, this is not always a monomorphism. For the underlying reduced scheme $X_{\text{red}}$ of $X$, the pushforward homomorphism $$ K_0(X_{\text{red}})\to K_0(X) $$ is an isomorphism (via devissage). If you read Manin's "Lectures on the K-functor", you will see that the natural map $$ \text{Pic}(X) \to K^0(X) $$ is an injection. Yet the composition to $K_0(X)$ factors through the pullback homomorphism $$ \text{Pic}(X)\to \text{Pic}(X_{\text{red}}), $$ which may easily have a kernel, cf. Hartshorne, Chapter III, Exercise 4.6.