Why it is worth the trouble
Density is a bulk property that does not care about colour, coatings or surface weathering. It is the fastest way to separate galena from a similar-looking sulfide, or to catch a glass imitation.
Once you have felt a specimen of galena, the phrase heavy for its size stops being vague.
The method
Weigh the dry specimen. Then suspend it fully in water from a thread, without touching the sides or bottom, and weigh again. Specific gravity is the dry weight divided by the difference between the two.
Use a clean, non-porous specimen. Anything that absorbs water, or that dissolves in it, will give a wrong answer or be damaged.
Limits
Porosity, inclusions and attached matrix all pull the result toward wrong. A specimen still on its host rock cannot be measured this way at all.
Expect agreement to within a few percent of published figures. Better than that from a kitchen scale is luck.