Hi Evgeniya

The best way to get pure mitochondria is to do a final sucrose gradient after isolating the heavy membrane fraction containing mitochondria. You can directly lyse them in 1x loading buffer for western blot and probe for mitochondrial markers (HSP60, CoxIV; VDAC) and cytosolic or cytoskeletal proteins (Tubulin, actin) for determining the purity.

Its is not very clear to me from your question that what kind of activity assays you would like to perform with the isolated mitochondria because mitochondria has diverse roles in biology including apoptosis and oxidative phosphorylation.

We frequently use isolated mitochondrias to determine effect of microbes or chemicals on mitochondrial apoptosis. For this we treat the cells with the reagent in question and then isolate the mitochondria and incubate them with recombinant pro-apoptotic proteins (Bax, Bim)  to check their sensitivity towards apoptosis.  

I guess you are interested in mitochondria function and dysfunction assays. There are multiple assays that can be done to test mitochondrial activity such as Proton Current, Proton Motive force, Change in Membrane potential, Respiratory control ratio etc. Each has its obvious positive and negative sides. Please go through the link to find a useful paper on the topic.

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3076726/

Thanks