My question is the following:
Is there a small complex, perhaps analogous to the Chevalley-Eilenberg complex, computing the (co)homology of a restricted Lie algebra over a field of characteristic $p>0$?
Even if the answer is no, I would appreciate any references dealing with such computations.
That was a large part of the subject of my 1964 PhD thesis. While the motivation came from algebraic topology, the relevant algebra was published separately in the paper http://www.math.uchicago.edu/~may/PAPERS/3.pdf. It is very obvious from the case of abelian restricted Lie algebras with zero restriction what the minimal size of a $V(L)$-projective resolution of the ground field $k$ can be, where $L$ is a restricted Lie algebra with enveloping algebra $V(L)$. Section 6 of that paper constructs a resolution $X(L)$ of that minimal size for any $L$, using the theory of twisted tensor products.
The serious part of the mathematics is to construct a coproduct on $X(L)$ suitable for computing $Ext_{V(L)}^*(k,k)$ as an algebra. In characteristic $2$, the coproduct is coassociative and very explicit, as explained in Remarks 10, op cit, and $X(L)$ embeds into the bar constuction via a map of differential coalgebras. In char $p>2$, the price one pays for minimal size is that the coproduct is defined inductively and is not coassociative. The embedding into the bar construction is then only compatible with coproducts up to homotopy. The paper also gives a spectral sequence for computing the restricted Lie algebra cohomology from the cohomology of the underlying Lie algebra; that eases calculations and minimizes the difficulty caused by the cited non-coassociativity.
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 Cohomology of Restricted Lie Algebras by Tyler Evans cited by Dietrich. This can be accessed at arXiv:math/0111090.
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.
There is a large literature on the cohomology of restricted Lie algebras in characteristic $p>0$, see the link to the PhD-thesis of Tyler J. Evans and the references given therein. Also the book "Representations of Algebraic groups" by J. C. Jantzen is a good reference, not only for the cohomology of restricted Lie algebras (although it is not in the references of Evans thesis).
As for the complex, just as taking $H^n(\mathfrak{g},M)=H^n(U(\mathfrak{g}),M)$
with the universal enveloping algebra $U(\mathfrak{g})$, one can use $H^n(\mathfrak{g},M)=H^n(u(\mathfrak{g}),M)$ for the restricted universal enveloping algebra, if $\mathfrak{g}$ is restricted. On the other hand,
for Lie algebras $\mathfrak{g}$ of a semisimple algebraic group $G$ in prime characteristic one can also consider the representation theory of $G_1$, the first Frobenius kernel of $G$. In fact, the representation theory of $G_1$ and $\mathfrak{g}$, considered as restricted Lie algebra, can be identified. In this setting the low-dimensional cohomology groups have been computed explicitly for minuscule representations (e.g., the first cohomology groups for classical Lie algebras by J. C. Jantzen, in Progress in Mathematics, Vol. 95, 1991).
