What the paper describes is what to do after a significant statistic is obtained. That's what you ask for! The procedure only use ranks.
Comparison After the H Statistic
When the obtained value of the H statistic is statistically significant, it indicates that at least one of the groups is different from the others. It does not indicate, however, which groups are different or whether the difference is meaningful, nor does it specify how many of the groups are different from each other. This next procedure, called "multiple comparisons between treatments," constructs pair-wise multiple comparisons to locate the source of significance.~ This procedure tests the null hypothesis that some groups u and v are the same against the alternate hypothesis that some groups u and v are different (Tab. 5). When the sample size is large, these differences are approximately normally distributed. Because there are a large number of differences and because the differences are not independent, however, the comparison procedure must be adjusted appropriately. An inequality is used, and the hypothesis of nodifference among the three groups is tested at the alpha level of significance of .05. The null hypothesis is rejected if the calculated difference among groups is greater than the critical difference. In this example, the difference between treatment groups 1 and 3 (ie, 9.33) is the only difference that is greater than the critical difference (ie, 7.38). It can therefore be concluded that treatment 3 led to a different, and in this case better, result because it provided a greater increase in the range of knee flexion than did treatment 1. (...)
And you should follow the procedure in table 5.You can even do it in a paper. You don't need SPSS.