Fused filament fabrication is a technology of additive manufacturing that uses molten thermoplastics for building parts. Due to the convenient shape of the raw material, a simple filament, the market offers a great variety of materials from simple to blends of compatible materials. However, finding a material with the desired properties can be difficult. Making it in-house or using a material manufacturer can be costly and time-consuming, especially when the optimum blend ratios are unknown or new design perspectives are tested. This paper presents an accessible method of producing core-shell filaments using material extrusion 3D printing. The printed filaments are characterised by a polycarbonate (PC) core and acryl butadiene styrene (ABS) shell with three material ratios. Their performance was investigated through printed samples. Additionally, the material mixing degree was studied by varying the extrusion temperature, nozzle feeding geometry, and layer thickness. The influence of all four factors was evaluated using a graphical representation of the main effects. The results showed that a core-shell filament can be processed using a 3D printer with a dual extrusion configuration and that the mechanical properties of the samples can be improved by varying the PC–ABS ratio. This research provides an accessible method for developing new hybrid filaments with a predesigned structure using a 3D printer.