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The incorporation of graphene nanoplatelets (GnPs) within a polymer matrix can play an important role in the physical properties and the functionality of the composite material. Composites consisting of low-density polyethylene (LDPE) and GnPs of different concentrations were developed by mixing GnPs with a molten form of the polymeric matrix. The effect of the GnPs content on the morphological, structural, and electrical properties of the composites were investigated. As shown, graphene presence and its concentration significantly modified the polymer matrix properties, a behavior that can be employed for tailoring its applicability in electrical applications. It was found that the increase of the graphene platelets concentration seems to promote the formation of graphene agglomerates, air gaps, and inhomogeneities, while higher dielectric constant/lower dielectric losses can be achieved.