In Web of Science you'll perform an author search and then create a citation report.
Click on the "+ more" tab and then choose Author Search
Enter your name and initial.
If you have a common name, it can be helpful to select the "Research Domain" and then "Institution" limits, but if you name is uncommon, you can select "Finish Search."
The records that are returned match the name entered. To see the H-index and other information for this set of records, click on the "Citation Report" link to the right and above the results.
The H-index for this set of records is displayed in the second box from the left.
You're not done! You need to delete all the papers by others with the same name as yours. Scoll down and start scanning the records displayed. If you see one that doesn't represent one of your publications, click the box to the left of the record. Then, just above the top-most record in the list, there is a "Go" button. Clicking this will remove the citation.
Continue this process until you reach the H-th citation (that is, the number that corresponds to your H-index, as currently displayed). You need not go further since records below H on the list do not affect the H-index.