Multi-coated filters are notoriously difficult to clean. The formulations of most lens cleaner fluids tend to smear on the materials coating the filter. Microfiber cloths are dangerous because they hold dust very well. While they won't deposit the dust already in the cloth on the filter, this can allow the dust trapped in the cloth to scratch the filter.
It doesn't sound like you have scratched the multi-coating of your filter yet, just smeared some cleaning fluid around on it. Here is what I do to clean multi coated filters:
- Remove the filter from the lens.
- Rinse under warm tap water. This should knock most of the loose dust off. By using warm water it will also help to break up body oils and other 'sticky' fluids a little.
- Dry with a clean lens tissue.
- Clean with Formula MC Lens cleaner. Put a few drops on a clean lens tissue or clean microfiber cloth and clean from the middle of the lens out in a circular motion. Repeat as needed until no residue is still visible on the lens.
- Dry the filter with another clean lens tissue.
Just as with the front element of lenses, it takes a lot of dust or dirt before performance is affected because the light rays are highly dispersed for any particular point source of light as they pass through the front of the lens, so try to avoid obsessively cleaning your multi-coated filter or lens every time they get a few specks of dust on them. Note: Don't wash the lens under running water, but otherwise you can clean your multi-coated front lens element this way as well. Just knock the dust off with a quality air blower or lens brush first.