Apologia
This answer was prepared in response to a previous question about Measles virus, but the question was removed for reasons I am not party to but which I can guess. As I spent some time on it I thought I’d post it in reply to this question. It is specific for measles virus, although there are obviously aspects that apply to other viruses, and deals with characterization as well as isolation. I am not a virologist (although I have done some biochemical work on pig herpes virus) so the it was is based solely on my reading of the papers cited and of Field’s Virology. I welcome suggestions for corrections or additions. It may seem to some that the question does not justify an answer of this detail. However it may also be of interest to young contemporary students and researchers who may have difficulty in envisaging the technological limitations of post-war experimental science.
Original Isolation and Characterization
The Edmonston strain of measles virus was isolated in 1954 by Enders and Peebles from the blood of a child with measles, David Edmonston. Before this scientists had found that an agent which was small enough to pass through filters that retained bacteria (i.e. possibly a virus) could cause measles when inoculated into monkeys and could be passaged in eggs containing developing chick embryos — the best method available until then. What Enders and Peebles managed to do was to maintain the virus in cells growing in tissue culture (a far more convenient host experimentally) — initially primary human kidney cells, and then monolayer cultures of chick embryo cells.
They used light microscopy to detect the changes in the cells that demonstrated that they were infected (the original paper contains many photographs of the cytopathogenic effects on infected cells). They could passage the virus by inoculating uninfected cells with infected cells, and they also isolated some virus from the cells by breaking open the cells and using the technique of centrifugation to separate the (larger) virus particles from the soluble cell contents and debris. (Subsequently much improved purification procedures were developed, resulting in preparations suitable for electron microscopy.)
One of the main techniques for characterizing viruses at this time was immunology. Thus, they found that the tissue culture fluid reacted with the antibodies in serum from measles patients by the complement fixation test, showing that it contained measles antigen. Previously they had showed that the cytopathogenicity of their preparations was neutralized by such antibody-containing serum.
Their paper ends with a summary of the evidence that what they isolated was the viral agent responsible for measles, and the further evidence needed for conclusive proof.
Subsequent characterization
The subsequent 60 years saw a steady improvement in analytical method, so that characterization did not rely solely on observation of cell pathology and immunology (the methodological repertoire of which also developed immeasurably). Electron microscopy in the 60s allowed cellular infection to be followed in greater detail and, after improvements in purification, the structure of the virus itself to be observed. The nucleic acid, protein and lipid envelope components could also be analysed with such purified virus.
The emerging techniques included gel electrophoresis of proteins (to determine their size and relative abundance, and to effect small-scale preparations). Larger scale purification of the virus proteins was possible with purified preparations, allowing amino acid sequencing, especially of the most abundant proteins (e.g. the nucleocapsid protein, N), and cDNA cloning and sequencing technology led to the sequencing of the whole Edmonston virus genome by Bellini et al. in 1985. This latter resulted in a much greater understanding of the gene expression of the virus, and provided tools such as specific oligonucleotide probes for various genes and antibody reagents to detect specific proteins.
Comment on Poster’s specific questions
The crux of the argument in the question would seem to be:
“If a researcher filters the particles in a serum…any sample filtered by size will potentially have hundreds of different viruses in it, all the same size range…”
The above addresses this only implicitly. To be explicit:
Using either the serum from patients infected with a specific virus, virus passaged in eggs or viruses grown in cell culture one is massively increasing the proportion of that virus to any contaminants.
It is not easy to grow viruses in cell culture. Therefore when conditions are found for one virus they are unlikely to be optimal for other (low-level) contaminants. Thus the actual passage in culture removes viral contaminants. The purification problem becomes one of separation from cell debris.
Modern Methods of Purifying Measles Virus
Below is a summary flow chart taken from the article Manufacture of Measles Viruses by Langfield et al..

One of the key methods in this proces, Tangential Flow Filtration, is also known as Cross-Flow Filtration and is described in this Wikipedia article.