cohomology - Classifying Algebra Extensions over a fixed extension?

I think your problem is not constrained enough to have an interesting answer. Notice that your intertwining condition can be rephrased by saying that $g: B to A$ is a homomorphism of $R$-algebras, where $A$ is given the structure of $R$-algebra given by $f$. In these terms, what you are looking for is the comma category $mathcal{C} = (mathbf{Alg}_R downarrow A)$, whose objects are precisely the pairs $(B, g)$ as above, and whose morphisms $operatorname{Hom}_{mathcal{C}}((B, g), (B', g'))$ are the $R$-algebra homomorphisms $h: B to B'$ such that $g = g' circ h$. I am not sure it is possible to capture this beast with a cohomology group of any sort.

What you can do is restrict the class of objects that you are looking at. For example, you can classify square-zero extensions: fixing an $A$-module $I$, you can look at $R$-algebras $B$ such that you have a short exact sequence $0 to I to B to A to 0$; the name square-zero comes from the fact that $I$ is an ideal of $B$ with $I^2 = 0$. You can read about them in the first chapter of Sernesi's Deformations of Algebraic Schemes.