How should vertical metallic blinds be cleaned?

remove the slats and wash them in the bathtub or outside

I wrap a rag around a pair of tongs and put over the slat and squeeze gently so the tongs grip the slat and pull. If they are super dirty use a damp clothe on the tongs. You could always take them down and hang them outside over a clothes drying line and wash them with a hose. Let them dry for several hours and rehang.

They do make a blind cleaning tool, but if they are really filthy as you say, that wouldn't be enough for me. I generally do my blinds in the bathtub. I take them out of their holder which is pretty easy and put in the tub for about 15 minutes in a solution of soapy or mr. clean solution-whatever you have on hand that is not too harmful. Then I rinse off with my hand sprayer of shower, and dry. If weather is permissive I will do outside. If you have metallic drying them is more important. So you can turn them one direction and hand dry them then the other. You can also use a hair dryer. From that point on you could start using a blind cleaning tool available on Amazon or Walmart etc.

Try wiping them down with a rag dipped in a solution of warm water and a little bit of a very mild detergent, ( a few drops to a gallon of warm to hot water). Use separate sponges to clean and rinse and a soft towel to wipe them dry. If you use that method, be sure to put something under the blinds to catch any water that may drip. OR you can buy some of those Mr. Clean magic sponges...simply wet it, squeeze out until damp and wipe away the dirt.

Magic erasers. You will need to vacuum after.