Let $f:A \to B$ be a homomorphism of Hopf algebras. I think that the subset $N:= \{a \in A \, |\, f(a) \in \mathbb{C}\cdot 1_B \, \}$ is a normal sub-Hopf algebra of $A$ (even though I am not sure if $f\otimes f \circ \Delta(a) \in \mathbb{C} 1_B \otimes \mathbb{C}1_B$ is sufficient for $\Delta(a) \in N\otimes N$..). If this is true, one can check that the ideal $AN^+$, where $N^+=ker(\epsilon_A|_{N})$, is in the kernel of $f$. Is it actually equal to $kerf$?
EDIT: Okay, this is clear, since $f(a)=0=0\cdot1_B$ implies $a\in N$, so we have $\epsilon_{A}|_N(a)=\epsilon_{B}\circ f(a)=0$, thus $a \in N^+$. In particular, we have $AN^+=N^+$. It remains to show that $N$ is a subcoalgebra of $A$..