Suppose that $C = \mathbb P^1$ and $U = \mathbb P^1\setminus \{0,\infty\}.$ Then
$\pi_1(U)$ is cyclic, freely generated by a loop around $0$.
The local system $L$ is thus given by the vector space $L_a$, equipped with an invertible operator,
call it $T$, corresponding to the action of the generator of $\pi_1(U)$. This operator $T$ is the monodromy matrix.
Now $H^1_c(U,L) = $ the space of $T$-invariants of $L_a$, while $H^2_c(U,L) = $ the space
of $T$-coinvariants of $L_a$. If you think of the possible Jordan decompositions of
$T$, and the fact that the trace is insensitive to the unipotent part, but just depends on
the semi-simple part, you'll see that the it's going to be hard to find any interesting relation of the type that you want. (E.g. if $L_a$ is $n$-dimensional, and $T$ acts by the
identity, or by a maximally non-trivial unipotent element, in both cases the trace of $T$ is equal to $n$, but in the first case the Betti numbers are also both equal to $n$,
while in the second, they are both equal to $1$.)
You might also wonder about the Euler characteristic $H^2_c(U,L) - H^1_c(U,L)$, but this is always
equal to (rank $L$) $\cdot \chi(U)$, and so is insensitive to the monodromy matrices.
Best Answer
Suppose $X$ is a connected CW complex with fundamental group $\pi:=\pi_1(X,x)$. Then the cellular chain complex $C_*(\widetilde{X})$ of the universal cover is a chain complex of free $\mathbb{Z}\pi$-modules, where $C_i(\widetilde{X})$ has rank equal to the number of $i$-cells of $X$. We can also regard $L$ as a $\mathbb{Z}\pi$-module. By definition $H^i(X;L)$ is the $i$-th homology of the cochain complex $\operatorname{Hom}_{\mathbb{Z}\pi}(C_*(\widetilde{X}),L)$. This can sometimes be computed directly, if the cell structure of $X$ and the boundary maps are well enough understood.
Then there are various tricks involving long exact sequences associated to short exact sequences of coefficient systems, transfer arguments, Shapiro's lemma, spectral sequences, and so on. For manifolds, there is Poincaré duality with local coefficients, which can sometimes inform computations. These methods are descirbed in Brown's "Cohomolgy of Groups" for $X=K(\pi,1)$, but mostly they apply more generally.
It really does depend on the space and the coefficient system as to which of these methods works best.