If you are having trouble painting your doll's eyes with heat-set or acrylic paints using a paint brush, you might want to try a paint pen or marker. The Sharpie product in this video is oil-based. Some alcohol-based art markers (like Prismacolor) will work too.
BUT, be sure to test any marker or paint pen on a scrap of baked clay first to ensure it is compatible with polymer clay. If it remains sticky or flakes, etc., it is not compatible. Also, if you are going to add a gloss over the paint or maker, be sure to test it too to be sure the gloss is compatible with the paint or marker ink.