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 $E_1^{i,j} = H^{i+j}(\text{gr }kG,\text{gr }M)_{(i)} \Rightarrow H^{i+j}(kG,M)$.