The magnetic and electrical properties of complex oxide thin films are closely related to the phase stability and cation ordering, which demands that we understand the process-structure-property relationships microscopically in functional materials research. Here we study multiferroic thin films of double-perovskite La2NiMnO6 epitaxially grown on SrTiO3, KTaO3, LaAlO3 and DyScO3 substrates by pulsed laser deposition. The effect of epitaxial strains imposed by the substrate on the microstructural properties of La2NiMnO6 has been systematically investigated by means of advanced electron microscopy. It is found that La2NiMnO6 films under tensile strain exhibit a monoclinic structure, while under compressive strain the crystal structure of La2NiMnO6 films is rhombohedral. In addition, by optimizing the film deposition conditions a long-range ordering of B-site cations in La2NiMnO6 films has been obtained in both monoclinic and rhombohedral phases. Our results not only provide a strategy for tailoring phase stability by strain engineering, but also shed light on tuning B-site ordering by controlling film growth temperature in double-perovskite La2NiMnO6 films.