To begin with, it is best to ventilate the room before cleaning. You should also have a mask for respiratory protection and goggles or glasses so that you do not damage your eyes from debris being released from the fireplace while cleaning.
I would recommend using a vacuum cleaner first as this will remove most of the dust and dirt particles if done correctly. However, if there are any stubborn grime parts then I think that using an old toothbrush can be useful because they allow good access into small crevices which may need extra attention in order to make sure everything gets cleaned properly.
Afterward, mix up some cement-based powder cleanser with water according to instructions on the packaging (should look like runny porridge). Then you can apply it to the fireplace and use an old paintbrush to work into grime areas. This should then be left for a few hours or overnight before having another go with a vacuum just to make sure that no residue is left behind.
More stubborn marks might require some gentle abrasive scouring using baking soda paste, but you need to ensure not to scrub too hard otherwise damage may occur. And if this does happen, there are specialist products that can help restore stone surfaces back to their original state more easily than trying to do so through normal household cleaning methods.
But please take care because these kinds of materials will scratch softer stones very quickly without people realizing what they’re doing until it’s already happened! So always try the test in small hidden areas before you go ahead and get to work on the whole fireplace.
Also, make sure that no abrasive materials come into contact with metal accessories such as decorative items or fire tools because these can be scratched too which will ruin their appearance. So if there are any areas of concern then it is best practice not to clean them until they are sorted out properly (with a professional).
Once everything has dried up completely, I would recommend applying some stone-specific oil/wax in order to seal porous surfaces so that future spills do not cause permanent damage. This should also help protect against water penetration when using the fireplace again in future winter months for heating purposes!