In general, the activity assay depends on the protein, but it will generally involve incubating the enzyme with it substrate and measuring the rate of disapearance of substrate/rate of appearance of product using a spectrophotometer. The activity assay you use will be the same at all points during the experiment, so you would use the same activity assay to measure the activity of your crude lysate and your column fractions. As your preparation becomes more pure, the specific activity of your sample (units of activty/mg of protein) should increase, reflecting the removal of contaminating proteins from your sample.