When $n$ is a prime, the additive structure of $H^*(BPU_n, \mathbb Z)$ has been computed independently in 

* Kameko, Masaki; Yagita, Nobuaki, _The Brown-Peterson cohomology of the classifying spaces of the projective unitary groups ${\rm PU}(p)$ and exceptional Lie groups_, Trans. Amer. Math. Soc. **360** (2008), no. 5, 2265–2284, doi:[10.1090/S0002-9947-07-04425-X](https://doi.org/10.1090/S0002-9947-07-04425-X)

and in  

* Vistoli, Angelo, _On the cohomology and the Chow ring of the classifying space of ${\rm PGL}_p$_, J. Reine Angew. Math. **610** (2007), 181–227, doi:[10.1515/CRELLE.2007.071](https://doi.org/10.1515/CRELLE.2007.071), arXiv:[math/0505052](https://arxiv.org/abs/math/0505052).

For $n = 3$, the second paper contains a computation of the multiplicative structure.