Let $k$ be a field. Given an affine algebraic group $G$ (defined as a functor from the category of $k$-algebras to the category of sets) then we have the coordinate ring (or the $k$-algebra representing $G$) $k[G]$.
Is there a standard way to put a co-algebra structure on $k[G]$? How can one explicitly define the maps $\Delta: k[G] \to k[G]\otimes k[G]$ and $\epsilon: k[G] \to k$?
On
This is maybe a bit indirect, but you can think of the functor of points of your affine algebraic group as factoring through the category of groups. Apply Yoneda's lemma to get yourself all the usual multiplication, inverse, and identity maps on the group, then the maps on the associated Hopf algebra are obtained by pulling back.
You can do this by identifying $k[G]$ with the set of morphisms from $G$ to $\mathbb{A}^1$. Similarly, identify $k[G]\otimes k[G]$ with morphisms from $G\times G$ to $\mathbb{A}^1$.
Now, we can use the group structure on $G$ to define the comultiplication as follows: Let $f\in k[G]$ and define $\Delta(f)$ to be the map that takes $(x,y)$ to $f(xy)$.
Similarly, you can do this to get the counit by using that $G$ has an identity element.
Note that in the above, I am abusing notation big time. Do tell me if you want more details with less abuse of notation.