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](https://arxiv.org/abs/1608.08054) and [a paper of Campos and Willwacher](https://arxiv.org/abs/1604.02043). 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](https://arxiv.org/abs/1604.02043), 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.