We report on the study of excited states in Kr populated via projectile Coulomb excitation at safe energies. The radioactive ion beams at energies of 2.85 MeV/u were delivered by the REX-ISOLDE facility at CERN and impinged on self-supporting Pt targets. The emitted γ-rays were detected by the Miniball detector-array. A detailed description of the experimental techniques used for extracting diagonal and transitional matrix elements and of the theoretical framework is given. The present experiment reveals the moderate evolution of the collective structure in the considered neutron-rich Kr isotopic chain, which is supported by the interacting boson model combined with the self-consistent mean-field method using a microscopic Gogny energy-density functional. The theory also suggests possible shape coexistence in the exotic nucleus Kr.