I would suggest you do not use shortening for a deep fryer. Its smoke point is too low. I recommend an oil such as peanut, corn or canola.
In a previous life, I worked in a fish and chip store. The only thing that will clean your oil is a good filter and cheesecloth is as good as any since it will take out the solids and let the oil flow through. The oil does have a lifespan and needs to be thrown out occasionally. We would use it for a week in the store and then change it up. For home use, you can probably use it 15-20 times perhaps and then change it. Once the oil gets too dark, the food has a burned flavour. As well, to keep the flavour, many people put a bit of the old oil into the new oil. Similar to the sourdough bread philosophy, to keep the taste.
Oooooooooo! Sketch, I'm going to come over there and bash you
"the food has a burned flavour." the oil has got to be gone before that happens
In a prevoius life I owned a fish and chip shop, I cooked in four 25ltr vats and worked them down the line, top up #1 from #2 and #2 from #3 and so on until #1 was worn out [7 days] Then fill #4 with clean oil and start over.
To clean your oil cheese cloth is OK but there is a material we call Bilene which does a professional job and is cheap as chips. This is the material taylors or dress makers use to stiffen their garments and it comes in three grades. The oil must be reasonably warm to get through the thinner grade and very hot to get through the thicker grade. Try and find this product as it is very good
My experience from when I was 19 could have been flawed, but I certainly hope we can still be friends?