Grime is formed when dust keeps accumulating on the grease previously applied to the hinges. Removing this will make it easier for you to lubricate the hinges and fix the squeaky door.
The best way to go about it is to spray ample amounts of WD-40 Flexible Straw on the hinges. All in all, it has excellent penetrating capabilities which will clean the grime from the outside. In addition, it will also take the grime out from all the concealed parts of the hinges.
Make sure you have an old rag ready to catch any drips and use the bendable straw for precision application to avoid getting it on nearby surfaces.