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Background: Neutron-deficient krypton isotopes are of particular interest due to the coexistence of oblate and prolate shapes in low-lying states and the transition of the ground state from one dominate shape to another as a function of neutron number. Moreover, the onset of large $E2$ transition strength around ${}^{76}$Kr indicates the erosion of the $N=40$ subshell gap.Purpose: A detailed interpretation of these phenomena in neutron-deficient Kr isotopes requires the use of a method going beyond a mean-field approach that permits the determination of spectra and transition probabilities. The aim of this work is to provide a systematic calculation of low-lying states in the even-even ${}^{68--86}$Kr isotopes and to understand the shape coexistence phenomenon and the onset of large collectivity around $N=40$ from beyond relativistic mean-field studies.Method: The starting point of our method is a set of relativistic mean-field plus BCS wave functions generated with a constraint on triaxial deformations $(\ensuremath{\beta},\ensuremath{\gamma})$. The excitation energies and electric multipole transition strengths of low-lying states are calculated by solving a five-dimensional collective Hamiltonian (5DCH) with parameters determined by the mean-field wave functions. To examine the role of triaxiality, a configuration mixing of both particle-number- and angular-momentum-projected axially deformed states is also carried out within the exact generator coordinate method based on the same energy density functional.Results: The energy surfaces, the excitation energies of ${0}_{2}^{+}$, ${2}_{1}^{+}$, and ${2}_{2}^{+}$ states, as well as the $E0$ and $E2$ transition strengths are compared with the results of similar 5DCH calculations but with parameters determined by the nonrelativistic mean-field wave functions, as well as with the available data. The results show a picture of oblate-triaxial-prolate shape transition. Coexistence of low-lying excited ${0}^{+}$ states is found to be a common feature in the neutron-deficient Kr isotopes. The underlying mechanism responsible for the shape coexistence is discussed.Conclusions: The main features of the low-spin spectra and the systematics of excitation energies and transition strengths in the neutron-deficient Kr isotopes are reproduced very well. The effects of dynamic correlations and triaxiality turn out to have important influences on the balance between the competing oblate and prolate states. An exact treatment of configuration mixing of particle-number- and angular-momentum-projected triaxial states is highly demanded to pin down these effects.