Is there a reasonable notion of an inner automorphism of a Hopf algebra $H$ which in the case of a group ring $H=\mathbb KG$ for a group $G$ reduces to a conjugation by $g\in G$?



Inner automorphisms of Hopf algebras
I am not sure if the following is the kind of answer you are expecting, but take the (left) adjoint action $(ad\_l h)\triangleright k=\sum h\_1 kS(h\_2)$ of a hopf algebra $H$ on itself.  

(It is known that the left adjoint action of $H$ on itself is always *inner* (in the sense of the def. 6.1.1, p. 87, in Montgomery's book "*Hopf algebras and their action on rings*") and turns $H$ into a $H$-module algebra).  

If you set $H=kG$, then, the left adjoint action for $g\in G$ reduces to action by conjugation: $(ad\_l g)\triangleright k=gkg^{-1}$.


More generally, for any *inner action* $\ \triangleright : H\otimes A\rightarrow A$, it is easy to show that any grouplike element $g\in G(H)$, acts as an inner automorphism of $A$. In case $H=A=kG$ and $\triangleright$ is the left adjoint action then this inner automorphism is the action by conjugation (of the corresponding group element). 


Concluding, i would say that the notion of an "*[inner action](http://www.ams.org/journals/tran/1986-298-02/S0002-9947-1986-0860387-X/S0002-9947-1986-0860387-X.pdf)*" (p.674-675) may be the reasonable notion you are looking for. 


