When traveling abroad, I would use a purifier instead of a filter. Either chemicals or a First Need. Reason being, when abroad I would have more concern about viruses in addition to bacteria or protozoa. The backcountry sources I usually tap into are primarily going to carry more risk for the latter issues than for viruses. For urban use, I would consider a Steri Pen - ultraviolet light would do the job and you could easily get batteries for it or replace the unit should you have to.

Coffee filters or a bandanna will strain out large particles like leaf matter. If by 'really dirty' you mean full of silt - you may want to look at adding a flocculant to the kit. Some contaminants can be settled out of water if you have that kind of time. A flocculant can speed up the process.