Water purification is a two step process:
1. Filtration to remove large dirt particles
2. Sterilisation to remove pathogens from the visibly clean looking water produced by filtration.
You can't skip the first step. Particles of dirt aren't just unpleasant to drink, they can also make the sterilisation step less effective by enclosing pathogens so the sterilisation agent can't reach them.
So if you're in a dire emergency you can use lots of layers of cotton cloth or other natural fibres for the filtration step, but if you have the option I'd recommend buying one of these: http://millbankbags.co.uk/ which is a dense fabric bag that filters water under gravity.
You want to keep careful track of which container has dirty water in and which has filtered water in - it sounds obvious but it's surprising how many highly intelligent people you have to tell at in fieldwork situations for using the same container for the dirty and filtered water.
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