Exact sequences of groups and Tannakian formalism By work of Deligne and others (I am following Deligne-Milne's notes which I just began to read: http://www.jmilne.org/math/xnotes/tc.pdf) we know that a given affine group scheme G can be recovered from their category of representations Rep(G) (thought of as a neutral Tannakian category). If G-->G' is a morphism Prop. 2.21 characterizes when is it faithfully flat (resp. closed immersion) in terms of properties of the induced map Rep(G')-->Rep(G).
My question is the following: is it possible to characterize that the sequence K-->G-->G' is exact in terms of the induced maps Rep(G')-->Rep(G)-->Rep(K).
( Actually I am interested in a situation when Rep(G')-->Rep(G) is known and one wonders if one can "construct" Rep(K) ).
 A: About your main question I suggest looking at Appendix A in
On Nori's Fundamental Group Scheme
Hélène Esnault, Phùng Hô Hai, Xiaotao Sun
Geometry and dynamics of groups and spaces, 377–398, Progr. Math., 265, Birkhäuser, Basel, 2008. 
http://arxiv.org/abs/math/0605645
http://link.springer.com/chapter/10.1007%2F978-3-7643-8608-5_8
I quote : 
"Let $L\xrightarrow q G\xrightarrow p A$ be a sequence of
homomorphism of affine group scheme over a field $k$. It induces a
sequence of functors
${Rep(A)\xrightarrow{p^*}Rep(G)\xrightarrow{q^*}Rep(L)}$
where $Rep$ denotes the category of finite dimensional
representations over $k$.
Theorem
With the above settings we have
(i) the map $p:G\to A$ is faithfully flat (and in particular
surjective) if and only if $p^*$ is a full subcategory of $Rep(G)$, closed
under taking subquotients.
(ii) the map $q:L\to G$ is a closed immersion if and only if any object of
$Rep(L)$ is a subquotient of object of the form $q^*(V)$ for some $ V\in
Rep(G)$.
(iii) Assume that $q$ is a closed immersion and that $p$ is
faithfully flat. Then the sequence $L\xrightarrow q G\xrightarrow
p A$ is exact if and only if the following conditions are fulfilled: 
(a) For an object $V\in Rep(G)$, $q^*(V)$ in $Rep(L)$ is trivial if and only if $V\cong p^*U$ for some $U\in Rep(A)$.
(b) Let $W_0$ be the maximal trivial subobject of $q^*(V)$ in $Rep(L)$. Then there exists $V_0\subset V$ in $Rep(G)$, such that $q^*(V_0)\cong W_0$.
(c) Any $W$ in $Rep(L)$ is embeddable (hence by taking duals a quotient) of $q^*(V)$ for some $V\in Rep(G)$.
The statements (i) and  (ii) are due to Deligne-Milne. "
About your original interest you could have a look at :
Milne, J. S.
Quotients of Tannakian categories.
Theory Appl. Categ. 18 (2007), No. 21, 654–664. 
http://arxiv.org/abs/math/0508479
From the introduction :
"Given a tannakian category $\mathsf{T}$ and a tannakian subcategory
$\mathsf{S}$, we ask whether there exists a quotient of $\mathsf{T}$ by
$\mathsf{S}$, by which we mean an exact tensor functor $q\colon\mathsf{T}%
\rightarrow\mathsf{Q}$ from $\mathsf{T}$ to a tannakian category $\mathsf{Q}$
such that


*

*the objects of $\mathsf{T}$ that become trivial in $\mathsf{Q}$ (i.e.,
isomorphic to a direct sum of copies of $1$ in $\mathsf{Q}$) are precisely
those in $\mathsf{S}$, and

*every object of $\mathsf{Q}$ is a subquotient of an object in the image
of $q$.
When $\mathsf{T}$ is the category $Rep(G)$ of finite-dimensional
representations of an affine group scheme $G$ the answer is obvious: there
exists a unique normal subgroup $H$ of $G$ such that the objects of
$\mathsf{S}$ are the representations on which $H$ acts trivially, and there
exists a canonical functor $q$ satisfying (a) and (b), namely, the restriction
functor $Rep(G)\rightarrow Rep(H)$ corresponding to the inclusion
$H\hookrightarrow G$. By contrast, in the general case, there need not exist a
quotient, and when there does there will usually not be a canonical one. In
fact, we prove that there exists a $q$ satisfying (a) and (b) if and only if
$\mathsf{S}$ is neutral, in which case the $q$ are classified by the
$k$-valued fibre functors on $\mathsf{S}$. Here $k=End(1)$ is assumed to be a field."
A: You can think about category $Rep(K)$ as about $G-$equivariant sheaves on $G'$. This translates into the following: let $A$ be the algebra of functions on $G'$; this is
commutative algebra in the category $Rep(G')$. Using fully faithful functor $Rep(G')\to Rep(G)$ we can consider $A$ as a commutative algebra in category $Rep(G)$. Now $Rep(K)$
is equivalent to $A-$modules in the category $Rep(G)$ as a tensor category. Equivalently,
$Rep(K)$ is de-equivariantization of $Rep(G)$ with respect to $Rep(G')$.
A: Given two elements $V,W$ of $Rep_G$, we can compute $Hom_{K} (V, W)$. It is just the maximal subobject of $V^{\vee} \otimes W$ that comes by pullback from a representation of $G'$. (Because it is the maximal $K$-invariant subrepresentation of that)
I think $Rep_{K}$ is essentially the universal abelian category containing those objects and those morphisms. If $K$ is reductive, I can make this expicit - you start with the representations of $G$, give them their new endomorphism algebras, and then split all the idempotents to get the representations of $K$. So that is an explicit description.
But I don't know exactly what to do about unipotents.
