The influence of the Dzyaloshinskii-Moriya interaction (DMI) on stripe domains in perpendicularly magnetized thin ferromagnetic films is theoretically and experimentally investigated. We develop a domain spacing model describing the dependence of the stripe domain width on the magnetic properties of the sample. By including the magnetostatic energy of the domain walls the model correctly describes the transition from Bloch to Neel walls with increasing DMI constant. An approach to determine the magnitude of the DMI constant by fitting the stripe domain width as a function of the effective perpendicular anisotropy of wedge-shaped samples is developed and applied to several ultrathin multilayer samples based on Ni/Fe/Cu(001). The magnitude of the DMI constant arising from Fe/Ni and Ni/Fe interfaces is 0.3 +/- 0.14 meV/atom, indicating that the domain walls are in a pure chiral Neel state. Furthermore, phase diagrams of the skyrmionic bubble domain phase are recorded for two samples with different DMI constants, and by scaling the magnetic field a universal phase diagram for perpendicularly magnetized systems is obtained.