A representation of G on a vector space V is a descent datum for V, viewed as a vector bundle over a point, to BG. That is, linear representations of G are "the same" as vector bundles on BG. So the question is equivalent to the analogous question about deformations of vector bundles on BG. We could just as easily ask about deformations of vector bundles on any space X.
Given a vector bundle V on X, consider the category of all first-order deformations of V. An object is a vector bundle over X', where X' is an infinitesimal thickening (in the example, one may take X = BG x E where E is a local Artin ring and X' = BG x E' where E' is a square-zero extension whose ideal is isomorphic as a module to the residue field). A morphism is a morphism of vector bundles on X' that induces the identity morphism on V over X.
If X is allowed to vary, this category varies contravariantly with X. Vector bundles satisfy fppf descent, so this forms a fppf stack over X.
This stack is very special: locally it has a section (fppf locally a deformation exists) and any two sections are locally isomorphic. It is therefore a gerbe. Moreover, the isomorphism group between any two deformations of V is canonically a torsor under the group End(V) (this is fun to check).
Gerbes banded by an abelian group H are classified by H^2(X,H) (this is also fun to check); the class is zero if and only if the gerbe has a section. If the gerbe has a section, the isomorphism classes of sections form a torsor under H^1(X,H). The isomorphisms between any two sections form a torsor under H^0(X,H). (This implies that the automorphism group of any section is H^0(X,H).)
In our case, H = End(V), so we obtain a class in H^2(X,End(V)) and if this class is zero, our gerbe has a section, i.e., a deformation exists. In this case, all deformations form a torsor under H^1(X,End(V)), and the automorphism group of a deformation is H^0(X,End(V)).
All of the cohomology groups above are sheaf cohomology in the fppf topology. If you are using a different definition of group cohomology, there is still something to check.
Best Answer
You can look at Manetti's paper Deformation theory via differential graded Lie algebras, arXiv:math/0507284.
As the title suggest, it follows the philosophy that every deformation problem is governed by a DGLA, via solution of Maurer-cartan equations (module gauge action).
One of the main advantages with respect to the classical Kodaira-Spencer's approach is that the theory works over any field $\mathbb{K}$ of characteristic $0$.
The construction of the Kuranishi map is presented in Section $4$.