There are three resources you can use to count citations per year for journal articles.
1) Web of Science: Use an 'Author Search' to locate all articles by a particular author and then use the 'citation report' link to see total citations by year.
2) Scopus: Use the 'Author Search' link to locate all articles by a particular author and then select 'view citation overview' to see total citations by year.
3) Google Scholar: This service is only really useful if the author has set up a public profile. If they have, you can use the public profile to view citations by year for all identified publications. Otherwise, a search on a single publication will provide you with a citation count however it is important to check the linked citations to make sure they are legitimate citations as they do not go through any quality control checks as in the other databases.
Each resource has it's own strengths and limitations, and you may find that no single one covers all of the author's publications. To get the most comprehensive idea of citations (and eventually h-index), it may be necessary to look at the citations in each resource and compile the results manually.
Your Faculty Liaison Librarian can help you with the use of these resources and the calculation of bibliometrics. https://library.sydney.edu.au/contacts/subjectcontacts.html