criteria can be chemical (BOD, oxygen, nutrients; salinity; toxic substances, ...) hydrological, morphological or ecological (macrozoobenthos, fish, water plants, algae). Based on earlier methods like Saprobic index (benthic organisms indicating for BOD and O2) and river morphology and motivated by the European Water Framework Directive (EU-WFD) many different methods have been developed, especially during the last few years, see summary:
Can you be more specific with regards to why you need to classify the river? A lot of the river classifications are somewhat skewed towards temperate climates and practically all classifications have an empirical bias of sorts. Some are heavily focussed on the physical aspects... It would be useful to know the rationale behind the classification, as some are more suited than others.
The EU-waterframework directive (WFD) imposes a monitoring of a.o. physical characteristics of rivers. This link provides to the benchmark report: http://www.eea.europa.eu/publications/92-9167-001-4 . I was always impressed form the UK and the German method.