The H-index in Web of Science
When you have created a list of publications of the researcher you want to investigate, you can create a Citation Report, based on the records in Web of Science.
Use the link Create Citation Report at the right side on the top of the results.
Here you can see the H-index of the researcher, the number of citations and citing articles, also without self-citations. This information is based on the articles in the results list, so you have to make sure that list is accurate and up-to-date (otherwise the H-index could for example be partially based on the articles and citation received by someone else!).
What does 'without self-citations' mean in Web of Science? This is the total numer of citing articles minus any article that appears in the set of search results on the Citation Report. This means this number is not determined by matching author names: it is possible you see articles from the author in the list of citing articles. In that case, check your search criteria, maybe you excluded certain author sets.