One can view $A$ and $B$ as sitting completely isometrically inside their injective envelopes $I(A)$ and $I(B)$. Then by injectivity a unital *-homomorphism (or more generally a unital completely positive map) $f:A\rightarrow B\subseteq I(B)$ extends to a unital completely positive map $\overline f:I(A) \rightarrow I(B)$.

I've been digging around but I am not convinced that this $\overline f$ will necessarily be a homomorphism or unique. Hopefully this spurs on the great brains to more fully answer your question.