Filters should be cleaned at least once a month and preferably every 2 weeks if you can. However, you do not clean new filters until they are at least 6 weeks old otherwise you can wash the filter bacteria out and mess up the biological side of the filter. Generally once a filter is 2 months (8 weeks old) you can start cleaning it every 2-4 weeks.
When you do clean a filter you squeeze/ rinse out filter materials in a bucket of tank water. When they are clean put them in the aquarium and wash the filter case and motor out under tap water. When the filter case and motor have been cleaned you reassemble the filter and put the filter media back in, fill it with tank water and turn it back on.
*NB* Make sure your hands and the plug are dry before plugging it back in.
If you have filter cartridges, you can make a small incision along the top or bottom and tip out the granules that are in the cartridge and throw the granules away. Then squeeze the cartridge out in a bucket of tank water and re-use it.
You can normally add sponges to most external filters and these give you more filtration area. You simply get a square or rectangular sponge for any brand of filter and use a pair of scissors to cut the sponge so it fits in the filter.
You can get round/ cylindrical sponges from some brands of internal power filter. These cylindrical sponges have a hole through the middle of them and they usually fit over the intake strainer of most external filters. They add more filtration area and reduce the chance of the fish being sucked onto the intake strainer.
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If the eye is cloudy it is infected. However, doing a few big water changes will normally fix the problem unless it becomes badly infected. Then the eye will stick out a lot more than normal, it can go red and they sometimes pop. Normally tho you will see it go cloudy and start to bulge out. It's quite noticeable.
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Fish are fine with water changes. Get yourself a basic model gravel cleaner like the one in the following link. Use it to drain some water out and clean the gunk out of the gravel. Leave the fish in the tank when you do this and just slowly move around the tank cleaning the gunk out and draining water. When you have removed about 75% of the water, stop cleaning and refill the tank with dechlorinated water.
https://www.about-goldfish.com/aquarium-cleaning.html
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If algae is causing a film on the surface it will be green. Having live plants in the tank will utilise excess light and reduce the chance of algae growing. Water Sprite (Ceratopteris thalictroides/ cornuta) is one of the better plants for Bettas. It floats on the surface but can also be grown in the substrate.
If someone is using hair spray, deodorant, perfume, smoking or anything else that releases gasses into the air, that can leave a film on the water. Lack of surface turbulence can also cause it, as can dust.
Make sure you don't have any moisturising creams or anything on your hands when you feed the fish or work in the tank.
If you notice it just do a water change and it should get rid of it. If it continues to happen regularly, then post a pic of it and see if it has an oily sheen to it when you look at it. If your not sure what I mean by an oily sheen, add a couple of drops of vegetable oil to a glass of water and look at the surface. it should get a bluey sheen across the surface.
If the fish ever looks off colour or you have concerns the water looking odd or smelling bad, generally the best thing to do is a water change. It will dilute anything in the water and usually fixes the problem.