I'm not sure how best to answer the question formulated here, but I can comment further on references. As Dietrich says, there is a large literature. Ever since the foundational work by Jacobson and Hochschild, the study of restricted Lie algebras and their cohomology has become fairly widespread, but also fragmented. There is some literature dealing with fairly arbitrary restricted Lie algebras, which usually doesn't go too far: for instance, the U.C. Davis thesis by Tyler Evans cited by Dietrich. This can be accessed here.
Though arXiv captures only some of the more recent work, you can find a sample there by searching for "cohomology of restricted Lie algebra" or the like. This already makes it clear that the subject goes in many directions, ranging from algebraic topology (as in work of Peter May and others) to algebraic groups, etc.
Special cases tend to lead to more explicit methods and computations, which are easiest to track down using MathSciNet if you have access. At one extreme would be the solvable restricted Lie algebras, at the other extreme the simple ones. The latter include those of Cartan type, not directly related to algebraic groups, along with those close to the Lie algebras of simple algebraic groups. For instance, Jantzen's large book Representations of Algebraic Groups (2nd ed., AMS, 2003) focuses on this last topic. Here the restricted Lie algebra has the same hyperalgebra and cohomology as the first Frobenius kernel, so the study of higher Frobenius kernels is also natural. In another direction, the study of reduced enveloping algebras (analogous to the finite dimensional restricted enveloping algebras) raises further possibilities.
References abound, but I'll mention just a few other names to search for among those currently active in the field: Eric Friedlander, Brian Parshall, Daniel Nakano, Cornelius Pillen, Chris Bendel, Julia Pevtsova, Jorg Feldvoss, Chris Drupieski.