I think that a general example is the so-called Larson's character, which in a sense ties together the trace and determinant functions.
To make the long story short: Let $C$ be a cocommutative bialgebra, $V$ a vector space and $EV$, the exterior algebra. Then, it has been shown that:
If $C\otimes V\rightarrow V$ is an action which makes $V$ a $C$-module, then there is a unique measuring $EV\otimes V\rightarrow EV$, extending the action on $V$.
In this sense, $EV$ becomes a $C$-module, with $C\cdot E^kV\subset E^kV$. If we furthermore assume that $\dim V=n$ then $E^nV$ is 1-dim. Let it be spanned by $\{z\}$. For any $c\in C$, let $\chi(c)$ be defined by $c\cdot z=\chi(c)z$. In this way, a linear map $\chi:C\rightarrow k$ is defined. It can be easily shown that this is an algebra map. It is called the Larson's character.
It can furthermore be shown that, if $g$ is a grouplike element of $C$ then $\chi(g)=\det T_g$, where $T_g:V\rightarrow V$ is explicitly given by $v\mapsto g\cdot v$; and that if $g$ is a primitive element then $\chi(g)=Trace(T_g)$.
For a detailed presentation of the above, you can see ch. VII, sect. 7.1, p.146-153, from Sweedler's book on Hopf algebras.
Furthermore, you can also take a look at Larson's paper on Characters of Hopf algebras. However, the presentation there looks quite different:
Larson adopts a dual point of view (to the usual notion of characters in group/algebra representation theory) and develops a theory of characters based on comodules of Hopf algebras. He actually considers characters as elements of the hopf algebra (instead of functionals on it) which are associated with comodules over the hopf algebra rather than modules over the hopf agebra. Although i have not studied his paper in detail, i think that his approach is essentially equivalent to the one followed in Sweedler's book; the usual notion (as the one presented above) can be recovered if we adopt Larson's approach and start from comodules of the finite dual $C^{\circ}$ hopf algebra.