Shameless promotion ahead. If you are interested in cohomology with base field $\mathbb{R}$, then:
- For simply connected manifolds without boundary you have my paper and a paper of Campos and Willwacher. Both papers give so-called "real models" for the configuration spaces; these models are commutative differential graded algebras, and in particular the cohomology of the model is the cohomology of the configuration space.
- For simply connected manifolds with boundary (and interiors of such manifolds, it's the same thing), we have a paper with Campos, Lambrechts, and Willwacher, where we also give a real model (several, in fact). If $\dim M \ge 4$ then this model is fairly explicit, otherwise it's complicated.
Besides all the models we give are equipped with a symmetric group action that models the natural symmetric group action on $\operatorname{Conf}_k(M)$, so if you are interested in unordered configuration spaces then you can compute their cohomology by considering invariants.