If normal cleaning of the eyepiece doesn’t change anything, the dirt is likely inside the viewfinder path rather than on an outer surface. In that case, simple swabbing won’t fix it.

For a Nikon D5300, the practical DIY option is partial disassembly of the camera/viewfinder area and cleaning the internal optical parts. Community answers point to teardown and repair guides for the D5300 and closely related Nikon bodies (such as the D5000/D5100/D3200) that show how to access the viewfinder assembly.

Be aware this is not a quick job: expect a fairly involved procedure, roughly around 30 minutes, with risk of damaging small parts if you’re inexperienced. If you do attempt it, use a proper blower and lens-safe cleaning methods, and avoid introducing liquid into the camera.

So the short answer is: if the debris is internal, cleaning it usually requires opening the camera and following a teardown/repair guide for the D5300 or a very similar model.