It varies from one researcher to the another, but most of citation databases (including Web of Science) provide both kind of h-indices and you can compare them to get a clearer profile of the specific author.
First of all, it is important to specify that you cannot completely avoid self-citations. They have their distinct role - they present your previous results, and avoid repetition of parts that you have already written in previous contributions. Under these circumstances, they save a lot of space and help you avoiding plagiarism.
Many researchers consider about 15% self-citations out of the total number of citations is normal. If this number is exceeded by several units, it is not dramatic. But when the percentage is exceeding 20-25%, you should analyse very seriously why it happens. Are you over-citing yourself? Are the members of the international community sufficiently interested in your work? Is the topic you develop in a particular situation (are there very few researchers working on this particular topic)? Of course, when the percentage of self-citations increases, the difference between the H-index calculated on all the citations and the H-index calculated after excluding the self-citations will increase.
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