How do I clean H-Alpha Filter?

You just don't. It isn't going to affect your images so let it be. Besides, that brush is too hard, you'd need a much softer one (and more expensive too).

In your case the filter looks good. To remove the dust you can use a blower that comes with camera lens cleaning kit.

Unless you have something like grease on the filter it's best not to clean it. But if you must then you can submerge the filter in a bawl of distilled water (not tab or mineral water) after adding like two drops of dish washing soap in the distilled water.  Leave it there for few hours then holding the filter from its sides shake it a bit in the water then remove it from the bawl and pure some distilled water on it to wash it from any residues on both sides then leave it standing on it's side to dry (not in a flat position so the water can slide away from the filter).

That's the safest way.

Again your filter looks good, Just a blower will do the job.

I clean my filter often.  Most filter manufactures, such as Astrodon, claim their coating are tuff enough to resist scratching during cleaning for numerous cleanings.   If they don't, don't buy that brand.   The particles you refer to are adhered to the surface  and will take water moistening to loosen them.  You do not want to use abrasive action to clean the filter surface such as rubbing.  Dry particles can scratch the coatings.   There are a number of lens cleaning products on the market and home made recipes.  Celestron use to recommend a homemade recipe to clean their optics.  Most will be a combination of mostly distilled water, 91% Isopropyl Alcohol and a vey small part of mild detergent.  I prefer Orion of California lens cleaner.  A very small drop on a COTTON "Q" tip will be best.  Apply the cleaner with one side, flip and gently rolling the Q Tip between your fingers, start to dry the surface.  the surface will fog up.  Throw the 1st Q Tip away.  Using another Q Tip repeat rolling the Q Tip stim between you fingers.  The fogging will take some time to completely remove. Flip the Q Tip and continue till completely dry.  Now blow off and cotton fibers with a Bulb blower.  Screw  the filter into the carousel.  Blow all dust off of every filter, are clean if necessary.  Be sure to examine both side of the filter carousel and blow all dust off, then put the carousel in to the holder and close quickly.

If one does this properly and does not expose the filter wheel opening to dust contamination, taking new flats should not be necessary for each session. 

I use a similar process on my $8K AP130GTX  and Tak FSQ106ED.  Except I use 100% cotton balls.  If you live in a highly wooded area, as I do, and image in the spring.  You had better damn well clean the pollen off you optics.  That stuff will adhere to the surface.   And some trees put out sap droplets.

Lynn Krizan

Astrodon (actually Farpoint) has a YouTube video on how to clean their filters.

https://youtu.be/t5NAIbV1l8o?si=lOvZyeWW730bVfvx