Water Purification for SHTF scenarios comes down to pretty much Three methods: Distillation (The Gold Standard); Filtration; and Chemical Treatments.
Distillation for SHTF means that you probably will have/need an Atmospheric Pressure Distiller. In its simplest form, that is a Boiling Chamber/Retort that has a single outlet that can be piped to a Condenser/Heat Exchanger, from which flows the Purified Water. Boiling Water takes a lot of Energy, and takes a while to accomplish. Using Glass or Stainless Steel for the Retort Chamber is best, but Copper and Cast Iron work nearly as well. Same for the Condenser. I have listed the materials in order, Best First. Skills required to make a Distiller and the materials required make this a fairly difficult task after SHTF.
Distillation eliminates nearly all impurities. With a relatively simple apparatus, you can end with Water that is chemically very pure, without Suspended Solids, and containing no Biological matter/organisms, and free from Heavy Metals. Water that has been distilled has no taste, and no smell. It can be a little "Flat" tasting if warm and not Aerated. Distilled Water is Great for Batteries, Radiators, and People.
Filtration is the Next best strategy for Water Purification, and when combined with Chemical Treatment, it is nearly as good as Distillation at removing Pathogens and Impurities. It is a faster method and is less Energy hungry. This is the method that will probably be used/selected by the highest number of Survivors. Building a Filtration system is a lower skill-required task, and uses simpler and more easily obtained materials than Distillation. After SHTF, it will be easier to build a Filtration Rig than a Distiller. Filtration and Chemical treatment together can provide Water with no measureable Biological matter, little suspended solids, but it is not very good at eliminating some deleterious chemical compounds nor Heavy Metals. Filtered Water can taste great, but may not. A "Special Case" of Filtration is Reverse Osmosis. It can make water that challenges Distillation for Purity, and there are People-Powered systems available - Cheaper Than Dirt sometimes sells the Swedish Hand-Pump system, for about $700. I would Highly recommend this if you want Portable and Sufficient for more than a Couple. Neatest is that "Him Who Wants It" can supply the energy to make the water! Other RO systems rely on supply water pressure over 60 PSI in order to move a reasonable amount of water through the RO membrane. Disadvantages of RO are that the membrane can be penetrated, or Plugged, and are difficult to replace without poisoning the system.
Chemical Treatment is Fastest, and can work in a matter of Minutes. You can process a small amount of water in a Canteen, while on the Run. It can poison you, if you don't follow directions. It kills, but does a poor job of removing Biological Materials from the Water. It does not remove any Heavy Metals. Generally, Chemically Treated Water tastes bad/flat, at best, and can also be stinky or off-putting to drink. Chemical Treatment implies a source of Chemicals - you have to Recognize, Obtain, Haul, Preserve, and use them carefully.