Relationship between the cohomology of a group and the cohomology of its associated Lie algebra

Yes, there is a connection. The cohomology of the Lie algebra is connected to the cohomology of the group via a spectral sequence.

I'm going to assume $k$ is a field of characteristic $p \geq 0$. Then it is a result of Lazard (Sur les groupes nilpotents et les anneaux de Lie, Ann. Sci. Ecole Norm. Sup. (3), 1954, 71, 101-190) that your Lie algebra L is a $p$-restricted Lie algebra over the field $\mathbb{F}_p$, where $\mathbb{F}_p=\mathbb{Z}$ if $p=0$. If $p=0$, then $L$ is a Lie ring over the integers.

Now let $I$ be the augmentation ideal of the group ring $kG$. We can filter the group ring by the powers of $I$, and get the associated graded ring $\text{gr } kG = \bigoplus_{n=0}^\infty I^n/I^{n+1}$. The associated graded ring inherits from $kG$ the structure of a Hopf algebra. It is a result of Quillen (On the associated graded ring of a group ring, J. Algebra, 1968, 10, 411-418) that $\text{gr } kG$ is isomorphic as a Hopf algebra to $u(L \otimes_{\mathbb{Z}} k)$, the $p$-restricted enveloping algebra of $L \otimes_{\mathbb{Z}} k$. (If $p=0$, then it is just the usual universal enveloping algebra, I think.)

Now, there is a spectral sequence connecting the cohomology of the associated graded ring $\text{gr } kG$ to that of $kG$: $E_1^{i,j} = H^{i+j}(\text{gr }kG,k)_{(i)} \Rightarrow H^{i+j}(kG,k)$. For the construction of this spectral sequence, you can see Section 3 of the paper Complexity for modules over finite Chevalley groups and classical Lie algebras by Lin and Nakano (Invent. Math., 1999, 138 (1), 85-101). That paper also contains some applications in the special case when $G$ is a finite group of Lie type of a certain kind, or is the $p$-Sylow subgroup of such.

Addendum: This last bit is something of an attempt to address Bugs Bunny's comment. Given a $kG$-module $M$, we can form the associated graded module $\text{gr }M = \bigoplus_{n=0}^\infty (I^n.M)/(I^{n+1}.M)$. Then $\text{gr }M$ is a graded $\text{gr }kG$-module, so by restriction a module for $L \otimes_{\mathbb{Z}} k$. Then you get a spectral sequence looking like $E_1^{i,j} = H^{i+j}(\text{gr }kG,\text{gr }M)_{(i)} \Rightarrow H^{i+j}(kG,M)$.

The continuous cohomology of a group $\Gamma$ is the direct limit $$H^*_{\text{cts}}(\Gamma;\mathbb Q)=\lim_{\longrightarrow}\ H^*(\Gamma/K;\mathbb Q)$$ of the cohomology of all its finitely generated nilpotent quotients $\Gamma/K$. The basic properties of continuous cohomology are established in Hain, "Algebraic cycles and extensions of variations of mixed Hodge structure", 175–221 in Complex geometry and Lie theory, Proc. Sympos. Pure Math 53.

There is an obvious comparison map $H^*_{\text{cts}}(\Gamma;\mathbb Q)\to H^*(\Gamma;\mathbb Q)$, which is always an isomorphism on $H^0$ and $H^1$, and is always injective on $H^2$. A finitely generated group $\Gamma$ is called pseudo-nilpotent if this map is an isomorphism in every degree.

Nomizu's theorem implies that for finitely generated groups, $H^*_{\text{cts}}(\Gamma;\mathbb Q)$ coincides with the continuous cohomology $H^*_{\text{cts}}(\mathfrak{g};\mathbb Q)$ of the Malcev Lie algebra $\mathfrak{g}$ of $\Gamma$. The Malcev Lie algebra is a certain pronilpotent $\mathbb Q$–Lie algebra associated to $\Gamma$ mentioned above by Tom Goodwillie. It has the property that its associated graded $\text{gr}(\mathfrak{g})$ is isomorphic to your $\hat{L}=\text{gr}(\Gamma)$, the associated graded of the group $\Gamma$. Moreover, in many (possibly all?) cases there is an isomorphism $H^*_{\text{cts}}(\mathfrak{g};\mathbb Q)\approx H^*(\hat L;\mathbb Q)$.

Some examples of pseudo-nilpotent groups—that is, groups for which the cohomology of the group and its associated Lie algebra coincide—are free groups, fundamental groups of Riemann surfaces, and pure braid groups. (This property is closely related to the property of a space $X$ being a rational $K(\pi,1)$, in the sense that the localization-at-0/rationalization of $X$ is aspherical.) Definitely not all groups have this property, however. For example, the reason that the pure braid group is pseudo-nilpotent is that it is the fundamental group of the complement of a particularly nice hyperplane arrangement. But without some condition on the arrangement, Falk showed that there are aspherical-hyperplane-complement-groups that are not pseudo-nilpotent.

I recommend reading Hain's excellent article for more information; unfortunately I could never find it online, though you can read snippets on Google Books. In the interest of full disclosure: much of this answer was taken from my paper "Representation theory and homological stability", with Benson Farb (arXiv:1008.1368, pages 61-62).

The following two papers seem to go, independently, more or less in line with Tom Goodwillie's comment above:

• Yu.V. Kuz'min, On the connection between group cohomology and Lie algebras, Russ. Math. Surv. 37 (1982), N4, 123-124 http://dx.doi.org/10.1070/RM1982v037n04ABEH003950 .
• P.F. Pickel, Rational cohomology of nilpotent groups and Lie algebras, Comm. Algebra 6 (1978), N4, 409-419 http://dx.doi.org/10.1080/00927877808822253