Does there exist a simple characterization of epimorphisms between affine group schemes over a field ? Are they faithfully flat morphisms ?
There are papers by Bien and Borel on epimorphisms in the category of algebraic groups over a closed field. According to these papers, the answer to your question is no: there is no simple characterization. A necessary criterion for an immersion $H\hookrightarrow G$ to be an epimorphism is that all regular functions on $G/H$ are constant. So $SO(n)$ in $SL(n)$ is not epimorphic but a Borel subgroup in any connected group is.
Edit: I just looked it up. The criterion above is also sufficient.
Today Brion posted a paper titled Epimorphic subgroups of algebraic groups on the arXiv.
The theorems in this paper directly answer your question for all fields (not just algebraically closed fields), generalizing the aforementioned results of Bien and Borel.
In particular, quoting from the abstract:
"We show that the epimorphic subgroups of an algebraic group are exactly the pull-backs of the epimorphic subgroups of its affinization. We also obtain epimorphicity criteria for subgroups of affine algebraic groups, which generalize a result of Bien and Borel."